1,1/2,2/3,3/4,4/5,5/6
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0,1,2,3,4,5,6,7,8,9,10
ãšãªããã®ãçºèŠããããå ±åããŠã»ããã
No.2223GAI2024幎10æ8æ¥ 07:41
äŸãã°ãïŒïŒïŒïŒïŒ/ïŒïŒïŒ/ïŒïŒïŒ/ïŒããšããããšã§ããïŒïŒïŒã
1=1
2=1/(1/2)
3=1/((1/2)*(2/3))
4=1/((1/2)*(2/3)*(3/4))
5=1/((1/2)*(2/3)*(3/4)*(4/5))
6=1/((1/2)*(2/3)*(3/4)*(4/5)*(5/6))
No.2224管çè
2024幎10æ8æ¥ 09:04 åºæ¥ããå
šéšã䜿ã£ãŠã»ããã
No.2225GAI2024幎10æ8æ¥ 09:24
(1+2/3)/(4/5-3/4)/(5/6-1/2) = 100
(1+1/2*2/3*3/4*4/5)*5/6 = 1
1+(1/2+2/3-3/4+5/6)*4/5 = 2
(1+1/2+2/3+3/4+5/6)*4/5 = 3
(1+1/2)/(3/4*4/5)+2/3+5/6 = 4
1+(1/2-2/3)/((3/4-4/5)*5/6) = 5
1+(1/2+2/3)/(5/6-3/4*4/5) = 6
1-2/3+(5/6-1/2)/(4/5-3/4) = 7
(1-1/2*2/3)/(3/4-4/5*5/6) = 8
(1+1/2)/(2/3-3/4*4/5*5/6) = 9
(1-1/2)/(3/4+4/5-2/3-5/6) = 10
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šéšäœããããã§ãã
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No.2226ãããã2024幎10æ8æ¥ 12:24
ïŒãäœãã®ãã©ããã£ãŠãäœããŸããã§ãããèŠäºã§ãã
ããã°ã©ã çã«ããããšè©Šã¿ããã§ãããããŸãã«ãã¿ãŒã³ãå€å²ã«æž¡ãã®ã§ã»ãã®äžéšã®éšåã§ããå©çšã§ããŸããã§ããã
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No.2227GAI2024幎10æ8æ¥ 13:05
1ããªããŠã100ãå«ããŠãã¹ãŠäœããŸããã
(1/2+2/3-3/4+5/6)*4/5 = 1
(2/3+5/6)/(1/2+3/4)+4/5 = 2
(3/4-2/3)/(5/6-4/5)+1/2 = 3
(2/3-1/2)/((4/5-3/4)*5/6) = 4
(1/2+2/3)/(5/6-3/4*4/5) = 5
(5/6-1/2)/(4/5-3/4)-2/3 = 6
(5/6)/((2/3-1/2)*4/5)+3/4 = 7
(1/2*2/3)/((4/5-3/4)*5/6) = 8
(1/2*5/6)/(4/5-3/4)+2/3 = 9
(5/6-2/3*1/2)/(4/5-3/4) = 10
(5/6)/((2/3-1/2)*(4/5-3/4)) = 100
(è¿œèš)
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No.2228ãããã2024幎10æ8æ¥ 15:16
å¥è§£ã§ããã
5/6*(1/(2/3-1/2))/(4/5-3/4)=100
((1/(3/4))-((2/3)/(1/2)))*(5/6)*(4/5)=0
(3/4-2/3)/(5/6-4/5)-1/2-1=1
((3/4-2/3)/(5/6-4/5)-1/2)*1=2
((2/3-1/2)/(5/6-4/5)-1)*(3/4)=3
((1-1/2)/(3/4-2/3))*(5/6)*(4/5)=4
((2/3-1/2)/(4/5-3/4))/(5/6)+1=5
(((((1/(1/2))/(2/3))/(3/4))/(4/5))/(5/6))=6
((3/4-2/3)/(5/6-4/5)+1)/(1/2)=7
((2/3-1/2)/(5/6-4/5)+1)/(3/4)=8
(3/4-1/2)/(5/6-4/5)+1/(2/3)=9
(1/(5/6+2/3+1/2))/(4/5-3/4)=10
No.2229kuiperbelt2024幎10æ8æ¥ 23:20
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(âãåæ°,#ãååæŒç®)ã®ãã¿ãŒã³ã ãã«éå®ããŠèª¿ã¹ãŠã¿ãŸããã
1/2+((5/6/4/5)/(2/3-3/4)) = 0
1/2*((3/4-2/3)/(5/6/4/5)) = 1
2/3/((1/2*3/4)-(5/6/4/5)) = 2
1/2+((2/3-3/4)/(4/5-5/6)) = 3
1/2/((3/4-2/3)/(4/5*5/6)) = 4
1/2/((2/3+5/6)*(4/5/3/4)) = 5
1/2/((2/3/3/4)/(4/5*5/6)) = 6
3/4-((1/2-2/3)/(4/5/5/6)) = 7
2/3/((1/2/3/4)+(5/6/4/5)) = 8
1/2/((3/4-2/3)*(4/5*5/6)) = 9
1/2/((3/4-2/3)+(4/5-5/6)) =10
ããããããã®çµæã䜿ãããŠè²°ã£ãŠ
5/6/((2/3-1/2)*(4/5-3/4)) =100
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šéšãå¶èŠã§ããŸããã
No.2230GAI2024幎10æ9æ¥ 08:56
ãã®åŒã§ã¯ã1/2,2/3,3/4,4/5,5/6ã䜿ã£ãŠãããããšã«ã¯ãªããªãã®ã§ã¯ïŒ
äŸãã°0ã®åŒã®äžã®5/6/4/5ã¯åŒãéåžžéãã«è§£éããŠ
5÷6÷4÷5=5/(6Ã4Ã5)
ãšèšç®ããã°ç¢ºãã«0ã«ãªããŸããã
(5/6)/(4/5)ãªãã°åŒã®å€ã¯-12ã«ãªããŸãã
ãã¹ãŠã®åæ°ã«ã«ãã³ãè£ã£ãŠèšç®ãçŽããš
(1/2)+(((5/6)/(4/5))/((2/3)-(3/4))) = -12
(1/2)*(((3/4)-(2/3))/((5/6)/(4/5))) = 1/25
(2/3)/(((1/2)*(3/4))-((5/6)/(4/5))) = -1
(1/2)+(((2/3)-(3/4))/((4/5)-(5/6))) = 3
(1/2)/(((3/4)-(2/3))/((4/5)*(5/6))) = 4
(1/2)/(((2/3)+(5/6))*((4/5)/(3/4))) = 5/16
(1/2)/(((2/3)/(3/4))/((4/5)*(5/6))) = 3/8
(3/4)-(((1/2)-(2/3))/((4/5)/(5/6))) = 133/144
(2/3)/(((1/2)/(3/4))+((5/6)/(4/5))) = 16/41
(1/2)/(((3/4)-(2/3))*((4/5)*(5/6))) = 9
(1/2)/(((3/4)-(2/3))+((4/5)-(5/6))) = 10
(5/6)/(((2/3)-(1/2))*((4/5)-(3/4))) = 100
ãšãªããŸãã
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No.2231ãããã2024幎10æ9æ¥ 13:22
ãïœãããïŒ
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(2/3)/(((5/6)/(4/5))-((3/4)*(1/2)))ã=1
(2/3)+(((1/2)/(3/4))+((4/5)*(5/6))) =2
(1/2)+(((2/3)-(3/4))/((4/5)-(5/6))) =3
(1/2)/(((3/4)-(2/3))/((4/5)*(5/6))) =4
(1/2)*(((2/3)/(4/5))/((5/6)-(3/4))) =5
(1/2)/(((3/4)/(2/3))-((5/6)/(4/5))) =6
(3/4)-(((5/6)/(4/5))/((1/2)-(2/3))) =7
(1/2)*(((2/3)/(5/6))/((4/5)-(3/4))) =8
(1/2)/(((3/4)-(2/3))*((4/5)*(5/6))) =9
(1/2)/(((3/4)-(2/3))+((4/5)-(5/6))) =10
ããšããŸããã
No.2232GAI2024幎10æ9æ¥ 17:57
ããã®ã¹ã¬ããã«ã¯ãã以äžè¿ä¿¡ã§ããŸãããããšãªããŸããŠããããããªãæ°èŠã¹ã¬ããã§ãã
No.2173 DD++ãã 9æ16æ¥ 16:55 ã®ãæçš¿ã«ã€ããŠè¿ä¿¡ã§ãã
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function hammingDistance(str1, str2) {
// ããã³ã°è·é¢ãèšç®ããé¢æ°
let distance = 0;
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}
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for (let j = 0; j < originalSeq.length; j++) {
for (let k = j + 1; k < originalSeq.length; k++) {
const newSeq = originalSeq.split('');
newSeq[j] = newSeq[j] === '0' ? '1' : '0';
newSeq[k] = newSeq[k] === '0' ? '1' : '0';
const z = newSeq.join('');
let count = 0;
for (const seq of sequences) {
if (hammingDistance(z, seq) === 2) {
count++;
}
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const sequences = ['0000000', '0010111', '1001011', '1100101', '1110010', '0111001', '1011100', '0101110'];
const result = checkCondition(sequences);
console.log(result); // true or falseãåºåããã
=====
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â, ðâ, ðâ, ðâ, ðâ, ðâ
ðâ, ðâ, ðâ, ðâ, ðâ, ðâ, ðâ
ðâ, ðâ, ðÌ
â, ðÌ
â, ðâ, ðâ, ðâ
ðâ, ðâ, ðâ, ðâ, ðÌ
â, ðÌ
â, ðâ
a ãš b ãšã®é¢ä¿ã¯ãa ãš c, b ãš c ãšã®é¢ä¿ã«ããã®ãŸãŸéçšããããã§ãã
枬å®çµæã§ãã q ã«ã€ããŠãa, b, c ãåœé貚ã®åè£ã§ããå Žåã«ã¯ããããå転ã®é¢ä¿ã¯äžã«å°œããããšãšãªããŸãã
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///////////
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///////////
â 第äžæ®µéã§ã®æž¬å®ã«ã€ããŠ
i ã 1 ãã 7 ãŸã§ã®æ·»åãšããŠäœ¿ããŸãã
j ã 1 ãã 8 ãŸã§ã®æ·»åãšããŠäœ¿ããŸãã
7人ããæè¡è
ã«ã1 ãã 7 ãšååãã€ããŸããæ·»åãšããŠã¯ãã£ã±ã i ã䜿ããŸãã
éœæ§éå Pj ã以äžã®ããã«å®çŸ©ããŸãã
P1 = {2, 3, 5}
P2 = {3, 4, 6}
P3 = {4, 5, 7}
P4 = {5, 6, 1}
P5 = {6, 7, 2}
P6 = {7, 1, 3}
P7 = {1, 2, 4}
P8 = {1, 2, 3, 4, 5, 6, 7}
â»äœè«ã§ãã P1 ãã P7 ã¯ãFANOå¹³é¢ãšãªã£ãŠããŸããP8 ã¯äœèšãã®ã§ãã
é°æ§éå Nj ã以äžã®ããã«å®çŸ©ããŸãã
N1 = {7, 6, 4, 1}
N2 = {1, 7, 5, 2}
N3 = {2, 1, 6, 3}
N4 = {3, 2, 7, 4}
N5 = {4, 3, 1, 5}
N6 = {5, 4, 2, 6}
N7 = {6, 5, 3, 7}
N8 = {}
8 æã®é貚ã«ä»¥äžã®ããã«ååãã€ããŸãã
C{Pj, Nj}
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ã¯ãPj ã®èŠçŽ ã« i ãå«ããããªé貚 C{Pj, Nj} ãã¬ã€ã¬ãŒã«ãŠã³ã¿ãŒã§èšæž¬ããŸãã
âæ©èŠè¡š | 1 2 3 4 5 6 7 âæè¡è
C{P1, N1} | 0 1 1 0 1 0 0
C{P2, N2} | 0 0 1 1 0 1 0
C{P3, N3} | 0 0 0 1 1 0 1
C{P4, N4} | 1 0 0 0 1 1 0
C{P5, N5} | 0 1 0 0 0 1 1
C{P6, N6} | 1 0 1 0 0 0 1
C{P7, N7} | 1 1 0 1 0 0 0
C{P8, N8} | 1 1 1 1 1 1 1
â»ãã®æ©èŠè¡šã§ã¯ã1 ãç«ã£ãŠããé貚ãèšæž¬ããŸãã
éœæ§ã®ã¬ããŒããããæè¡è
ã®éåã T ãšåã¥ããŸãã
Tããåœé貚ã®ããããæ¢ããšããããšãšãªããŸãã
ããšãã° T={2,3,5} ãªãã°åœé貚㯠C{P1, N1} ãšããããšãšãªããŸãã
â èšæž¬çµæTã®è©äŸ¡ã«ã€ããŠ
ç°¡åãªãã®ããé ã«ã
â T = Pj ãšãªã j ããããšã
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T â Pj
ãšãªã j ã¯ïŒã€ãããŸãã(FANOå¹³é¢ã®æ§è³ªã§ã)
ïŒã€ããåœé貚ã®åè£ C{Pj, Nj} ã«ã€ããŠã¯ç¬¬äºæ®µéã§ã(ãã以äžã¯èª€ã£ãã¬ããŒããçºçããªããã) åŠçå¯èœãšãªããŸãã
â¥Tã®èŠçŽ æ°ã 5 ã®ãšã
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â»C{P8, N8} ã«ã€ããŠãéœæ§ãé°æ§ãšèª€ã£ãã¬ããŒããïŒéçºçããã±ãŒã¹ã§ãã
åœé貚ã®åè£ãšããŠ
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â»j=8 ãé€å€ããŠã® C{Pj, Nj} ã«ã€ããŠãé°æ§ãéœæ§ãšèª€ã£ãã¬ããŒããïŒéçºçããã±ãŒã¹ã§ãã
誀ã£ãã¬ããŒããåºããæè¡è
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ããã FANOå¹³é¢ã®æ§è³ªã§ãã
{m,n} â Pj ãªã C{Pj, Nj} 㯠ïŒåãããŸãã
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æ°ãåãçŽããŠã以äžã§ã¯ãAâB ãå·®éåã®è¡šèšãšããŸãã
ã
âŠTã®èŠçŽ æ°ã 3 ã®ãšã
(ãã ããâ ã®ã±ãŒã¹ã¯é€ããŸãã)
â»éœæ§ãé°æ§ã«ãã誀ã£ãã¬ããŒãïŒéãšãé°æ§ãéœæ§ã«ãã誀ã£ãã¬ããŒããïŒéãšãèšïŒæ¬ã®èª€ã£ãã¬ããŒããçºçããŠããã±ãŒã¹ã§ãã
話ã®éœåäžãT ã«åºã¥ããŠUãäœããŸããUãå
šäœéåã§ãªããŠãã¿ãŸããã
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ãããŠãåœé貚ã®C{Pj, Nj}ã«ã€ããŠ
Pj = {x, y, z}
Nj = {p, q, r, s}
ãšããŸãã
T = {p, y, z,}
U = {x, q, r, s}
ã枬å®ã®çµæãšããŠåŸãããŠããããšãšãªããŸãã
Tã®ïŒã€ã®èŠçŽ ã®ãã¡ãåœã¬ããŒããã²ãšã€ããã®ã§ããã®å¯èœæ§ã¯äžåºŠïŒéããããŸãã
åœç©ã倧æåã§æžããš
Pj = {X, y, z}
Pj = {x, Y, z}
Pj = {x, y, Z}
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Tã«å«ãŸããïŒã€ã®èŠçŽ ã®ãã¡ïŒã€ãéžã¶ãšãããããã«å¯Ÿå¿ããŠé貚ãã²ãšã€å®ãŸããšããããšãšãªããŸãã
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ã«æ瀺ãããŠããã ããæ©èŠè¡šã§
åã³ã0 ãš 1 ãšããããããªããããäžã§ãäžèšã«åæ²ãããŠããã ããŸãã
C{P1, N1} | 1 0 0 1 0 1 1 //C4
C{P2, N2} | 1 1 0 0 1 0 1 //C6
C{P3, N3} | 1 1 1 0 0 1 0 //C7
C{P4, N4} | 0 1 1 1 0 0 1 //C3
C{P5, N5} | 1 0 1 1 1 0 0 //C5
C{P6, N6} | 0 1 0 1 1 1 0 //C2
C{P7, N7} | 0 0 1 0 1 1 1 //C1
C{P8, N8} | 0 0 0 0 0 0 0 //C0
ãªããåè¡ã®å³ç«¯ã¯ãããããå§ãã説æã®éœåäžãåé貚ã«æ°ããååã¥ããããã®ã§ãã
ãããåã®æä»çè«çåã®èšå·ãšããŠãâãã䜿ãããšãšããŸãã
ããšãã°ã 0110 â 1010 = 1100 ã§ãã
1 †n â€7 ã«ã€ããŠ
C0 â Cn = Cn
ã¯ãèªæã§ããã
ããã¯ãªããã®ãã以äžã®ããã«ãªã£ãŠããããšã§ãã
C3 = C2 â C1
C5 = C4 â C1
C6 = C4 â C2
C7 = C4 â C2 â C1
ã€ãŸããC1,C2,C4 ããç¥ã£ãŠããã°ã
ä»ã«ã€ããŠã¯ãïŒé²æ°ã®ä»æãã«ãã£ãŠå²ãåºãããšããããšã«ãªããŸãã
ããã¯ç§ã«ãšã£ãŠã¯éèªæãªããšã§ãã®ã§
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No.2220Dengan kesaktian Indukmu2024幎10æ7æ¥ 16:46
OEIS ã®
A075931
List of codewords in binary lexicode with Hamming distance 5 written as decimal numbers.
ããã
0,31,227,252,805,826,966,985,1354,1365,1449,1462,1647,1648,1676,1683
ãŸã§ãå©çšããŠ
笊å·é· 11 ãæå°ããã³ã°è·é¢ 5 ã®ç¬Šå·ã®ãããåã以äžã®ããã«äœæããŸããã
"12 0 3 00 4 0000"ââèŠåºã
"00 0 0 00 0 0000",//0
"00 0 0 00 1 1111",//1
"00 0 1 11 0 0011",//2
"00 0 1 11 1 1100",//
"01 1 0 01 0 0101",//4
"01 1 0 01 1 1010",
"01 1 1 10 0 0110",
"01 1 1 10 1 1001",
"10 1 0 10 0 1010",//8
"10 1 0 10 1 0101",
"10 1 1 01 0 1001",
"10 1 1 01 1 0110",
"11 0 0 11 0 1111",
"11 0 0 11 1 0000",
"11 0 1 00 0 1100",
"11 0 1 00 1 0011",
èŠåºãã«ã€ããŠèª¬æããŸãã
"12 0 3 00 4 0000"
ã§ã1,2,3,4 ã¯ãããŒã¿ããã(4ããã)ãšããŠäœ¿ãããããäœçœ®ã§ããäžäœããæé ã«ããŠããŸãã
"12 0 3 00 4 0000"
"01 1 0 01 1 1010",
äžã®äŸã§ã¯ã0101 ãæå³ããŠããã10é²ã§ã¯ 5 ã§ãããããã確ãã« 0 ãªãªãžã³ã§ 5 çªç®ã®ãããåã§ããããšã«çæããŠé ããã°å¹žãã§ãã
äžã®äžèŠ§è¡šã¯ã4ãããã®ããŒã¿ãããã®ç¬Šå·èªãã11ãããã«ãšã³ã³ãŒããããã®ãšãªã£ãŠããŸãã
ååã®æçš¿ãNo.2220 ã§çºçããŠããæ©èš¶äžæè°ãªçŸè±¡ããäžã§ããããŠããã®ãã«ã€ããŠããã°ã©ã ã§æ€èšŒããŸãããšããããªãŒã±ãŒãšãªããŸããã
"00 0 0 00 1 1111",//1
"00 0 1 11 0 0011",//2
"01 1 0 01 0 0101",//4
"10 1 0 10 0 1010",//8
ãç¥ã£ãŠããã°ã
ä»ã®11åã®ç¬Šå·èªã¯ãæä»çè«çåã§ãã£ãŠèšç®ã§ããã®ã§ãã
ãããããèªæã ãããšãæãã«ãªããããããããã£ããããããããŸããã
ããããªããã§ããã
ãã® A075931 ã®æ°åã¯ã0 ããé ã« 1 ã¥ã€ãã«ãŠã³ãã¢ããããŠãããããŒãã«äžã«ãããŠãã(å
è¡ãã)å
šãŠã®æ°ãšããã³ã°è·é¢ã5以äžãšãªãæ°ãã¿ã€ããŠã¯ããŒãã«ã«è¿œå ããŠæºã蟌ãã§ããã ãã§ãäœã£ãŠãããã§ãã貪欲ã¢ã«ãŽãªãºã ã§ãã£ãŠã°ãªãŒãã£ã«ãæãæå³ã§ã¯æ±ãäœãæ¹ã
ãªã®ã«ã[2220]ã®æçš¿ã§è§Šããæ³åãããããããããA075931 ã§ãç§ãåæã«èšå®ãããèŠåºãããã笊å·èªã®å€ãçŽã«ç€ºããŠãããªããŠãäžæè°ã§äžæè°ã§ãªããªãã®ã§ãã
ããšãã°ãæå°ããã³ã°è·é¢ã 6 ã§ã笊å·é·ã 13 ã笊å·èªæ°ã 13 ã®ã次ã®ç¬Šå·ã«ã¯ãä»è©±é¡ã«ããŠãã[æ³å]ãç§ã¯ã¿ã€ããããŸããã
"0011101111101",
"1001110111110",
"0100111011111",
"1010011101111",
"1101001110111",
"1110100111011",
"1111010011101",
"1111101001110",
"0111110100111",
"1011111010011",
"1101111101001",
"1110111110100",
"0111011111010",
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No.2221Dengan kesaktian Indukmu2024幎10æ7æ¥ 17:34
2221 ã§ç§ã¯ç¡æåæµã«ããŒã¿ãããã®äœçœ®ã決ããŠããã®ã§ããã(ã€ãŸããã§ãã¯ãããã®äœçœ®ã決ããŠããã®ã§ãã)
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No.2222Dengan kesaktian Indukmu2024幎10æ7æ¥ 22:36
èŠçŽ nåãã°ã«ãŒãkåã«åå²ããæ¹æ³ã«ã¯ã
èŠçŽ (n-1)åãã°ã«ãŒã(k-1)åã«åå²ããnçªç®ã®èŠçŽ ãkçªç®ã®ã°ã«ãŒããšããŠè¿œå ããå Žåãšã
èŠçŽ (n-1)åãã°ã«ãŒãkåã«åå²ããnçªç®ã®èŠçŽ ãkåã®ã°ã«ãŒãã®ã©ããã«è¿œå ããå Žåãšãããã
èŠçŽ nåãã°ã«ãŒãkåã«åå²ããæ¹æ³ã®æ°ãS(n,k)ãšãããšã
S(n+1,k)=k*S(n,k)+S(n,k-1)
ãšãã挞ååŒã§è¡šãããšãã§ããŠãS(n,k)ã¯ã第2çš®ã¹ã¿ãŒãªã³ã°æ°ãšããŠç¥ãããŠããŸãã
第2çš®ã¹ã¿ãŒãªã³ã°æ°ã¯ãS(n,1)=1(nâ§1),S(n,n)=1(nâ§0),S(n,0)=0(nâ§1)ã§ã以äž
S(3,2)=3,S(4,2)=7,S(5,2)=15,S(6,2)=31,...
S(4,3)=6,S(5,3)=25,S(6,3)=90,...
S(5,4)=10,S(6,4)=65,...
S(6,5)=15,...
...
ãšç¶ããŸãã
æºæ°éŠã®éŠå³ã®ç·æ°ã¯5åã®èŠçŽ ã5å以äžã®ã°ã«ãŒãã«åå²ããæ¹æ³ã®æ°ãªã®ã§ã
S(5,1)+S(5,2)+S(5,3)+S(5,4)+S(5,5)=52ãšãªããŸãã
éŠæšã®çš®é¡ã3çš®é¡ã ãšãéŠå³ã®ç·æ°ã¯
S(3,1)+S(3,2)+S(3,3)=5ã§
ââ¬âãâââãâââãâââãâââ
âââãâââãâââãâââãâââ
âââïŒâââïŒâââïŒâââïŒâââ
ãšãªããŸãã
éŠæšã®çš®é¡ã4çš®é¡ã ãšãéŠå³ã®ç·æ°ã¯
S(4,1)+S(4,2)+S(4,3)+S(4,4)=15ã§
ââ¬â¬âãââ¬ââãââ¬ââãââââãâââ¬â
ââââãââââãââââãââââãââââ
ââââïŒââââïŒââââïŒââââïŒââââ
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éŠæšã®çš®é¡ã4çš®é¡ã®ãã®ã¯ç³»å³éŠãšãã£ãŠãéŠæšã®çš®é¡ã3çš®é¡ã®ãã®ã¯äžç·éŠãšããããã§ãã
ç³»å³éŠã®éŠå³ã«ããæºæ°éŠãšåæ§ã«éãä»ãããŠããŠãäžã®å³ã§ã¯ãå¿éæµã®å Žåã¯ã
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æã¯éŠæšã®çš®é¡ã6çš®é¡ä»¥äžã®ãã®ããã£ãããã§ããéŠæšã®çš®é¡ã6çš®é¡ã ãšãéŠå³ã®ç·æ°ã¯
S(6,1)+S(6,2)+S(6,3)+S(6,4)+S(6,5)+S(6,6)=203ãšãªããŸãã
ãªãããã®éŠå³ã®ç·æ°ã¯ãã«æ°ãšãã£ãŠã
https://oeis.org/A000110
ã«ãèŒã£ãŠããŸãã
No.2219kuiperbelt2024幎10æ6æ¥ 21:21
2ãåºãšããäºé²å¯Ÿæ°log_{2}(x)ãlb(x)ã§è¡šããšãlb(3),lb(5),lb(7),lb(11),lb(13)ã®é£åæ°å±éã¯ã
lb(3)=[1;1,1,2,2,3,1,5,2,23,...]
lb(5)=[2;3,9,2,2,4,6,2,1,1,3,1,18..]
lb(7)=[2;1,4,5,4,5,4,1,29,...]
lb(11)=[3;2,5,1,1,1,25,1,1,...]
lb(13)=[3;1,2,2,1,22,23,1,9,149,...]
ã§ãlb(13)ã[3;1,2,2,1,22,23,1,9]ã§è¿äŒŒãããšã
[3;1,2,2,1,22,23,1,9]-lb(13)=201130/54353-lb(13)
=-2.25897731584196266637728075571*10^-12
ãšãªã£ãŠã201130/54353ãlb(13)ã®è¿äŒŒå€ãšãããšèª€å·®ãå°ããã®ã§ãlb(3),lb(5),lb(7),lb(11)ã
54353åãããšã
54353*lb(3)=86147.4668016970019305550728204
54353*lb(5)=126203.757741412805693795471951
54353*lb(7)=152588.162078596956051793358299
54353*lb(11)=188030.486767793017766203979679
ãšãªã£ãŠãå°æ°éšåã«çç®ãããšã
0.4668016970019305550728204=[0;2,7,33,...]
0.757741412805693795471951=[0;1,3,7,...]
0.162078596956051793358299=[0;6,5,1,...]
0.486767793017766203979679=[0;2,18,2,...]
ãªã®ã§ããããããã1/2=[0;2],3/4=[0;1,3],1/6={0;6],1/2=[0;2]ã§è¿äŒŒãããšã
(86147+1/2)/54353-lb(3)=6.10790627896702020627740913203*10^-7
(126203+3/4)/54353-lb(5)=-1.42428436437616975547826382828*10^-7
(152588+1/6)/54353-lb(7)=8.44124466103963591405650940218*10^-8
(188030+1/2)/54353-lb(11)=2.43449432087167148461376935139*10^-7
ãšãªããŸããã(86147+1/2)/54353,(126203+3/4)/54353,(152588+1/6)/54353,(188030+1/2)/54353ãé£åæ°å±éãããšã
(86147+1/2)/54353=[1;1,1,2,2,3,1,5,2,2,1,2,1,12]
(126203+3/4)/54353=[2;3,9,2,2,4,4,1,1,6,1,1,2]
(152588+1/6)/54353=[2;1,4,5,4,5,5,1,4,1,1,2,3]
(188030+1/2)/54353=[3;2,5,1,1,1,25,1,4,1,1,4,2]
ããã
lb(3)ã§[1;1,1,2,2,3,1,5,2,...]ã
lb(5)ã§[2;3,9,2,2,4,...]ã
lb(7)ã§[2;1,4,5,4,5,...]ã
lb(11)ã§[3;2,5,1,1,1,25,1,...]
ãŸã§äžèŽããŠããŠãlb(3),lb(5),lb(7),lb(11),lb(13)ãåæ¯ã54353*12=652236ã®åæ°ã§æ¯èŒçããè¿äŒŒã§ãããã§ãã
No.2210kuiperbelt2024幎9æ30æ¥ 20:09
lb(3),lb(5),lb(7),lb(11),lb(13)ãéåžžã«é£åæ°ããã®æã¡åãåæ°ã§
åæ¯ã6æ¡ã§ãããã®ããšããš
lb(3)=301994/190537
lb(5)=227268/97879ã(6æ¡ã§ãããã®ã¯åããªãã£ãã)
lb(7)=1273419/453601
lb(11)=1444074/417431
lb(13)=201130/54353
ãèŠã€ããã
kuiperbeltæ°ã«ããæ§æã§ã¯
lb(3)=172295/108706=1033770/652236
lb(5)=504815/217412=1514445/652236
lb(7)=915529/326118=1831058/652236
lb(11)=376061/108706=2256366/652236
lb(13)=201130/54353=2413560/652236
ããã§åãã®æ¹ã®ç°ãªãåæ°ã652236ãžãšçµ±äžãããšããã°ããããã®ååã¯
lb(3)-->round(652236/190537*301994)=1033770
lb(5)-->round(652236/97879*227268)=1514445
lb(7)-->round(652236/453601*1273419)=1831058
lb(11)-->round(652236/417431*1444074)=2256366
lb(13)-->round(652236/54353*201130)=2413560
ãšççŸãªãã€ãªãããŸããã
No.2211GAI2024幎10æ1æ¥ 08:46
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No.2205ks2024幎9æ27æ¥ 19:21
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No.2206ãããã2024幎9æ28æ¥ 01:03
1ã«ã€ããŠãÏã¯2以äžã®æ°ã§ãããé£ç¶ããæŽæ°ã®åã§ã¯æžããŸãããïŒ
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No.2207DD++2024幎9æ28æ¥ 08:46
çæ§ããææããããšãããããŸãã
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No.2208ks2024幎9æ28æ¥ 19:12
ç§ã®æããåäŸãšãã¯å
šéšç¡èŠã§ããïŒ
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No.2209DD++2024幎9æ29æ¥ 08:11
å³ç«¯ã®æ°å€ã¯å2ã€ã®å·®ã®çµ¶å¯Ÿå€ã§ãã
äžè¬ã«8ã®åæ°ã®èªç¶æ°ã¯ããããè¡ãããã§ãã
16=42-15*sqrt(3)=>0.019237886466840597088304877411914495791
24=123-70*sqrt(2)=>0.0050506338833465838817893053211345001350
32=276-63*sqrt(15)=>0.0020491889327362337062798137088245173490
40=525-198*sqrt(6)=>0.0010309289307365569377532082335043901511
48=894-143*sqrt(35)=>0.00059101675490591287205430668876207606729
56=1407-780*sqrt(3)=>0.00037009627571104859185362541955378301120
64=2088-765*sqrt(7)=>0.00024703558819826626394846596577432967552
72=2961-1292*sqrt(5)=>0.00017307027171223934761999919110380885978
80=4050-1197*sqrt(11)=>0.00012594458638060942551420518803934464147
88=5379-906*sqrt(30)=>9.4500095344005671898144251386011596633 E-5
96=6972-575*sqrt(143)=>7.2716696137850341565187520446959431717 E-5
104=8853-1350*sqrt(42)=>5.7149388688195943961281204512114323297 E-5
112=11046-783*sqrt(195)=>4.5728919063980887084329367575848946092 E-5
120=13575-899*sqrt(224)=>3.7160906777440839558589688710914293688 E-5
128=16464-1023*sqrt(255)=>3.0607247824951139183948832040071781991 E-5
136=19737-13860*sqrt(2)=>2.5508902623608594282453584631070366255 E-5
144=23418-1295*sqrt(323)=>2.1483200147407576879212274060209928031 E-5
152=27531-8658*sqrt(10)=>1.8262171743553579692301522935038486629 E-5
160=32100-1599*sqrt(399)=>1.5654351913667168657434798770696208115 E-5
168=37149-3526*sqrt(110)=>1.3520456453081349107645400277878002460 E-5
176=42702-1935*sqrt(483)=>1.1757513052464834609167108976471561999 E-5
184=48783-8460*sqrt(33)=>1.0288277537663826978867857967869962110 E-5
192=55416-11515*sqrt(23)=>9.0540344785055006911665795110653250388 E-6
200=62625-9996*sqrt(39)=>8.0096115343544563630858145670082038073 E-6
208=70434-40545*sqrt(3)=>7.1198701339296880836444048825359176973 E-6
216=78867-5830*sqrt(182)=>6.3571982558417182849809728108097647648 E-6
224=87948-9405*sqrt(87)=>5.6996944965601575244751615724698325284 E-6
232=97701-6726*sqrt(210)=>5.1298361531682648788761153900878672073 E-6
240=108150-3599*sqrt(899)=>4.6334908721224596314615983300244219419 E-6
248=119319-30744*sqrt(15)=>4.1991752820486645490899063644723485097 E-6
256=131232-4095*sqrt(1023)=>3.8174932812674583700545221491687345189 E-6
264=143913-34840*sqrt(17)=>3.4807064442220806178631564440052527937 E-6
272=157386-4623*sqrt(1155)=>3.1824025866890528095006338533945508999 E-6
280=171675-29394*sqrt(34)=>2.9172379591251503174251310489710550738 E-6
288=186804-5183*sqrt(1295)=>2.6807351648308626097063006717446215453 E-6
296=202797-32850*sqrt(38)=>2.4691236092811802692280772010616819375 E-6
304=219678-5775*sqrt(1443)=>2.2792126687875382004922224433354285596 E-6
312=237471-24332*sqrt(95)=>2.1082902188077300200429255193613423229 E-6
320=256200-6399*sqrt(1599)=>1.9540409567059134673684620949682723996 E-6
328=275889-26892*sqrt(105)=>1.8144802784198278034278698655600337535 E-6
336=296562-7055*sqrt(1763)=>1.6879004543876111616534704993720122179 E-6
344=318243-14790*sqrt(462)=>1.5728265895810835886254647889304459244 E-6
352=340956-23229*sqrt(215)=>1.4679804112725554698240184211417630657 E-6
360=364725-16198*sqrt(506)=>1.3722503533570500336644416285396892947 E-6
368=389574-25389*sqrt(235)=>1.2846667317585679159772552841492182749 E-6
376=415527-35340*sqrt(138)=>1.2043810565329850268087220171954593041 E-6
384=442608-64505*sqrt(47)=>1.1306487210116121767818178054760610574 E-6
392=470841-192060*sqrt(6)=>1.0628144602296206119864992601618119558 E-6
400=500250-69993*sqrt(51)=>1.0003000900280090029710013420615528427 E-6
No.2194GAI2024幎9æ25æ¥ 08:03
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No.2196ãããã2024幎9æ25æ¥ 16:47
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> 8ã®åæ°ããå³èŸºãå°åºãã決ãŸã£ãææ³ããããšããããšã§ããããã
ããæ¬ãèªãã§ããŠ
m:=1/2*(sqrt(n+1)+sqrt(n-1))â
ãšçœ®ããšã
1/m=sqrt(n+1)-sqrt(n-1)â¡
m^2=1/2*(n+sqrt(n^2-1))â¢
n-m^2=1/(4*m^2)â£
ãæãç«ã¡
ãããã®çåŒãçµã¿åãããããšã§
1/(8*m^5*(m+sqrt(n))^2)=(m-sqrt(n))^2/(8*m^5*(m^2-n)^2)
=(m^2-2*m*sqrt(n)+n)/(m/2)
=2*(m-2*sqrt(n)+n/m)
=(2*n+1)*sqrt(n+1)-(2*n-1)*sqrt(n-1)-4*sqrt(n)(*)
ãªãçåŒãæç«ããããšã«æãã
巊蟺ãèŠãããšã§å³èŸºã®å€ã¯O(1/(n^(5/2)*n)=O(1/n^(3+1/2)ã®ãªãŒããŒã§ã»ãç¡èŠåºæ¥ãããšãã§ããã
ãã®å·§ã¿ãªåŒãèŠãŠn+1,n-1,nãããããå¹³æ¹æ ¹âãå€ããånã«å¯Ÿå¿ããŠ
(2*n+1)*sqrt(n+1)â(2*n-1)*sqrt(n-1)+4*sqrt(n)
(2*n-1)*sqrt(n-1)â(2*n+1)*sqrt(n+1)-4*sqrt(n)
4*sqrt(n)â(2*n+1)*sqrt(n+1)-(2*n-1)*sqrt(n-1)
ãšãã颚ã«å·ŠèŸºã®èªç¶æ°ãå³èŸºã®2ã€ã®ç¡çæ°ã§æ§æåºæ¥ããªãšæã£ãŠå
çšã®çåŒãã©ããäœã£ãŠãããŸããã
ãªãããã§ç¬¬ïŒçªç®ã®çåŒã䞡蟺平æ¹ããŠ
16*nâ8*n^3+10*n-2*(4*n^2-1)*sqrt(n^2-1)
÷2ãã
8*nâ4*n^3+5*n-(4*n^2-1)*sqrt(n^2-1)â€
ãªãåŒãå©çšããã°
8ã®åæ°ã®èªç¶æ°ãå³èŸºã®äžã€ã®ç¡çæ°ã§è¿äŒŒåºæ¥ãããšãå¯èœãšãªãã
ãã®â€ããæ§æããŠãããŸããã
ããããããã®ææã®æ§ã«
gp > forprime(p=2,100,for(n=1,1000000,if(frac(n*sqrt(p)) < 0.000001,print(n";"p))))
978122;3
902702;7
283009;17
566018;17
345777;19
254813;29
509626;29
528641;41
424802;43
829254;47
528896;53
951113;61
977001;89
594030;97
ã
gp > forprime(p=2,100,for(n=1,10000000,if(frac(n*sqrt(p)) > 0.9999999,print(n";"p))))
9369319;2
7865521;3
7465176;5
3096720;7
9504180;11
2298912;17
4508361;19
9016722;19
5412001;23
8193638;29
9600319;31
1311360;41
3697884;43
7395768;43
6191808;47
8142716;61
2874480;71
5748960;71
8623440;71
4684249;73
4121279;79
8242558;79
9005009;89
6377352;97
ãªãçŽ æ°ãæŽæ°ãçµã¿åãããã°(bãšpãäžã®çµã¿åããã«åã£ãŠããæå³)
a+b*sqrt(p)
ã¯aã調ç¯ããããšã§äžåã®ç¡çæ°ã§ãªããŒã§ãèªç¶æ°ãã¹ãŠã奜ããªè¿äŒŒç²ŸåºŠã§äœãåºãããšã¯å¯èœãšãªããã§ããã
ããã(*)ã®çåŒãäœãåºããã»ã³ã¹ã«ã»ã©ã»ã©æå¿ããŸããã
No.2197GAI2024幎9æ25æ¥ 20:39
ãªãã»ã©ããã¯ã8ã®åæ°ã«ã ã䜿ããåŒããã£ãã®ã§ãããããããããŸããã
No.2198ãããã2024幎9æ25æ¥ 23:12
GAI ãããäœããããã®ãå
šãããããªããã§ããã
16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒã§ãããã®ããã«æ±ãããŠããçç±ã¯ã©ãã«ãããã§ãïŒ
No.2199DD++2024幎9æ26æ¥ 16:05
> 16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒã§ãããã®ããã«æ±ãããŠããçç±ã¯ã©ãã«ãããã§ãïŒ
ãããããããžã®è¿äºã®äžã®
8*nâ4*n^3+5*n-(4*n^2-1)*sqrt(n^2-1)â€
ãªãåŒãã
n=2,7,26ã§çãŸããåŒã以äžã®åŒãšãªãæå³ã§ãã
16=42-15*sqrt(3)
56=1407-780*sqrt(3)
208=70434-40545*sqrt(3)
ãã®
16=42-15*sqrt(3) ãç¹å¥ãªè¿äŒŒåŒãšããã€ããã¯ãããŸããã
äžã®3ã€ã®åŒãã
sqrt(3)=(42-16)/15=26/15
sqrt(3)=(1407-56)/780=1351/780
sqrt(3)=(70434-208)/40545=70226/40545
çãçºçããŸãã(åã«æ²ç€ºããŠããäžããéžãã§ããã ãã§ãã)
ããã«sqrt(3)ã®é£åæ°ãéäžã§æã¡åã£ãŠåæ°ãšããŠããæé ãèªåã§ããããŠãããš
gp > contfracpnqn(contfrac(sqrt(3)),20)
%208 =
[1 2 5 7 19 26 71 97 265 362 989 1351 3691 5042 13775 18817 51409 70226 191861 262087 716035]
[1 1 3 4 11 15 41 56 153 209 571 780 2131 2911 7953 10864 29681 40545 110771 151316 413403]
ã®æ§ã«å³ã«è¡ãã«ãããã£ãŠããsqrt(3)ãžè¿äŒŒããŠãããŸãã
ãã®æµãã®äžã«
26/15,1351/780,70226/40545ãããŸããã16=42-15*sqrt(3)ãç¹å¥ã®åŒãšã®è§£é㯠"ã¯ãŠïŒ"
ãšæã£ãŠããŸããŸãã
No.2200GAI2024幎9æ26æ¥ 17:08
ãããå
·äœçãªæ¹ãããããšæã£ãŠ1ã€æœåºããã質åã®æå³ãäŒãããŸããã§ããã
5çªã®åŒã§ã8n=ãã®åœ¢ã«ããåŒã ãç¹å¥èŠããŠããã®ã¯ãªãã§ããïŒ
ãã®åŒãäœããšãã«ããã®åã«èšèŒãããåŒãå¹³æ¹ããŠæŽçããŠåŸããã
4*n^3-3*n-(4*n^2-1)*sqrt(n^2-1)â0
ã®äž¡èŸºã«çªç¶ 8n ãå ããŠã§ããã®ã 5 çªã§ãããã
ããã§äž¡èŸºã« 7n ã足ãã° 7 ã®åæ°ã®åŒãã§ããã§ããããã9n ã足ãã° 9 ã®åæ°ã®åŒãã§ããããããªãã§ããããïŒ
8n ãéžæããå¿
ç¶æ§ã¯ãªãã§ãããïŒ
No.2201DD++2024幎9æ26æ¥ 19:22
> 5çªã®åŒã§ã8n=ãã®åœ¢ã«ããåŒã ãç¹å¥èŠããŠããã®ã¯ãªãã§ããïŒ
ãã®åŒãäœããšãã«ããã®åã«èšèŒãããåŒãå¹³æ¹ããŠæŽçããŠåŸããã
4*n^3-3*n-(4*n^2-1)*sqrt(n^2-1)â0
ãšèšèŒãããŠããéšåã¯
8*nâ4*n^3+5*n-(4*n^2-1)*sqrt(n^2-1)
ããªã
3*nâ4*n^3-(4*n^2-1)*sqrt(n^2-1)
ãšããªãã®ãïŒ
ãšè§£éããŠããã§ããïŒ
çç±ã¯ããããããšãæ°ä»ããªãã£ãã§ãã
2ã§å²ããããããã§çµãããšæã蟌ã¿ããã®ãŸãŸã§æ°å€ã§ç¢ºèªã«è¡ã£ãŠããŸããã
ããããã¯
6â32-15*sqrt(3)
9â108-35*sqrt(8)=108-70*sqrt(2)
12â256-63*sqrt(15)

ã§æ²èŒããŠããããšã§ãããã
ãç²æ«æ§ã§ããã
No.2202GAI2024幎9æ26æ¥ 20:04
> 3*nâ4*n^3-(4*n^2-1)*sqrt(n^2-1)
ããã«èšãã°ã3n ãš 4n^3 ãå·Šå³ã«åããæå³ããªããšæããŸãã
n ã¯å
·äœçãªæŽæ°ã代å
¥ããããšãæ³å®ããŠãããã§ããããããæŽæ°ã®è¿äŒŒåŒã«æŽæ°ã足ãé
ããã£ãŠã¯æ矩ãèãã§ãã
äžæ¹ã§ããããéã«å¹³æ¹æ ¹ã®æçè¿äŒŒåŒãšããŠ
â(n^2-1) = (4n^3-3n)/(4n^2-1)
ãšããåŒã¯äœãã«äœ¿ãããã§ããã
No.2203DD++2024幎9æ26æ¥ 21:46
DD++ããããã®ã¢ããã€ã¹ãåããŠsqrt(n^2-1)â(4*n^3-3*n)/(4*n^2-1)
ã®æŽ»çšãèŠãŠã¿ãŸããã
æéã®é¢ä¿ã§ããã°ã©ã ã®ãŸãŸã®å§¿ã§ç³ãèš³ãããŸããã
gp > for(n=1,100,if(core(n^2-1)==3,\
print(n,";sqrt(3)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/3))))
2;sqrt(3)=26/15
7;sqrt(3)=1351/780
26;sqrt(3)=70226/40545
97;sqrt(3)=3650401/2107560
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(3)),24)
%227 =
[1 2 5 7 19 26 71 97 265 362 989 1351
3691 5042 13775 18817 51409 70226
191861 262087 716035 978122 2672279 3650401
9973081]
[1 1 3 4 11 15 41 56 153 209 571 780
2131 2911 7953 10864 29681 40545
110771 151316 413403 564719 1542841 2107560
5757961]
-----------------------------------------------------------
for(n=1,10000,if(core(n^2-1)==5,\
print(n,";sqrt(5)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/5))))
9;sqrt(5)=2889/1292
161;sqrt(5)=16692641/7465176
2889;sqrt(5)=96450076809/43133785636
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(5)),20)
%222 =
[2 9 38 161 682 2889 12238 51841 219602 930249 3940598 16692641
70711162 299537289 1268860318 5374978561 22768774562 96450076809
408569081798 1730726404001 7331474697802]
[1 4 17 72 305 1292 5473 23184 98209 416020 1762289 7465176
31622993 133957148 567451585 2403763488 10182505537 43133785636
182717648081 774004377960 3278735159921]
-------------------------------------------------------------
gp > for(n=1,10000,if(core(n^2-1)==6,\
print(n,";sqrt(6)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/6))))
5;sqrt(6)=485/198
49;sqrt(6)=470449/192060
485;sqrt(6)=456335045/186298002
4801;sqrt(6)=442644523201/180708869880
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(6)),24)
%226 =
[2 5 22 49 218 485 2158 4801 21362 47525 211462 470449
2093258 4656965 20721118 46099201 205117922 456335045
2030458102 4517251249 20099463098 44716177445 198964172878
442644523201 1969542265682]
[1 2 9 20 89 198 881 1960 8721 19402 86329 192060
854569 1901198 8459361 18819920 83739041 186298002
828931049 1844160100 8205571449 18255302998 81226783441
180708869880 804062262961]
-------------------------------------------------------------
gp > for(n=1,10000,if(core(n^2-1)==7,\
print(n,";sqrt(7)=",(4*n^3-3*n)/(4*n^2-1)/sqrtint((n^2-1)/7))))
8;sqrt(7)=2024/765
127;sqrt(7)=8193151/3096720
2024;sqrt(7)=33165873224/12535521795
ããã«å¯ŸããŠé£åæ°ããã®è¿äŒŒ
gp > contfracpnqn(contfrac(sqrt(7)),36)
%233 =
[2 3 5 8 37 45 82 127 590 717 1307 2024
9403 11427 20830 32257 149858 182115 331973
514088 2388325 2902413 5290738 8193151
38063342 46256493 84319835 130576328 606625147 737201475
1343826622 2081028097 9667939010 11748967107 21416906117 33165873224
154080399013]
[1 1 2 3 14 17 31 48 223 271 494 765
3554 4319 7873 12192 56641 68833 125474
194307 902702 1097009 1999711 3096720
14386591 17483311 31869902 49353213 229282754 278635967
507918721 786554688 3654137473 4440692161 8094829634 12535521795
58236916814]
è¿äŒŒã¹ããŒãã皌ããŸãã
No.2204GAI2024幎9æ27æ¥ 07:23
ããããã°ã©ã ãPARI/GPã§èµ°ãããŠãããšãããã©ãããŠãããã°ã©ã ãç¹å®ã®å€ã§ã¯
çµæããããåŸ
ã£ãŠãçµäºããããã®åå ãäžã€ãã€æœ°ããŠãããšãããªããšæã£ãŠãããªã
次ã®ãããªèšç®ãè¡ãããŠããããšãå€æããŸããã
ãã®æ§ãªããšã«ãªã£ãŠããŸãã®ã¯ãç§ã䜿ã£ãŠãããœããã«éãã®ã§ããããïŒ
çããã䜿ãããŠãããœããã§ã¯åŠäœãªãçµæãè¿ããŠããããæããŠæ¬²ããã
gp > for(n=1,20,print(n";"floor(log(10^n)/log(10))))
1;1
2;2
3;2
4;4
5;5
6;5
7;7
8;8
9;9
10;10
11;10
12;11
13;12
14;14
15;14
16;16
17;16
18;18
19;19
20;20
3,6,11,12,13,15,17ã§ãã¡ãã®ææãè£åãããŠããŸããŸããã
No.2183GAI2024幎9æ19æ¥ 20:58
æå
ã§ã¯åæ§ã§ããã
PARI ã¯ãã²ãšãã³å°æ°ãæ±ãããšã«ãªããšïŒé²æ°ã§å
éšè¡šçŸããã®ããªïŒããšæããŸãããããã£ãšã¿ãããã ãã§ã¯ããŸã説æã§ããªããããªïŒ
? for(n=1,20,print(n";"floor(log(10^n)/log(10))))
1;1
2;2
3;2
4;4
5;5
6;5
7;7
8;8
9;9
10;10
11;10
12;11
13;12
14;14
15;14
16;16
17;16
18;18
19;19
20;20
No.2184Dengan kesaktian Indukmu2024幎9æ20æ¥ 00:05
JavaScript ã§ã¯ä»¥äžã®éãã§ãã
for (let n = 1; n <= 20; n++) {
console.log(n + ";" + Math.floor(Math.log(Math.pow(10, n)) / Math.log(10)));
}
äžã RUN ãããš
"1;1"
"2;2"
"3;2"
"4;4"
"5;5"
"6;5"
"7;7"
"8;8"
"9;8"
"10;10"
"11;11"
"12;11"
"13;12"
"14;14"
"15;14"
"16;16"
"17;17"
"18;17"
"19;19"
"20;20"
ãšãªããŸãã
JavaScript ã§ã¯ãã¶ãå°æ°ç¹ä»¥äžã¯ãæå¯ãã®ïŒé²æ°ã§æããŠããã®ã§âŠâŠ
No.2185Dengan kesaktian Indukmu2024幎9æ20æ¥ 00:13
ä»ã®èšç®ãœããã§ã調æ»ããŠã¿ãã
ïŒsageMathã®ãœãã
sage: for i in range(21) :print(i,floor(ln(10^i)/ln(10)));
(0, 0)
(1, 1)
(2, 2)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
(7, 7)
(8, 8)
(9, 9)
(10, 10)
(11, 11)
(12, 12)
(13, 13)
(14, 14)
(15, 15)
(16, 16)
(17, 17)
(18, 18)
(19, 19)
(20, 20)
å
šéšäžæãèµ°ã
ïŒRubyã®ãœãã
irb(main):001:0> include Math
=> Object
irb(main):012:0> 0.upto(20){|i| print i,";",log(10**i)/log(10),"\n"}
0;0.0
1;1.0
2;2.0
3;2.9999999999999996
4;4.0
5;5.0
6;5.999999999999999
7;7.0
8;8.0
9;8.999999999999998
10;10.0
11;11.0
12;11.999999999999998
13;12.999999999999998
14;14.0
15;14.999999999999998
16;16.0
17;17.0
18;17.999999999999996
19;19.0
20;20.0
3,6,9,12,13,15,18ã§äžæããããªããªãã
ïŒMaximaã®ãœãã
(%i13) for i :1 thru 20 do
print(float(log(10^i)/log(10)));
1.0" "
2.0" "
3.0" "
4.0" "
5.0" "
6.0" "
7.0" "
8.0" "
8.999999999999998" "
9.999999999999998" "
11.0" "
12.0" "
13.0" "
14.0" "
15.0" "
16.0" "
17.0" "
18.0" "
19.0" "
20.0" "
9,10ã§é£ç¹
No.2186GAI2024幎9æ20æ¥ 07:58
PARI ã§ãåºé¢æ°ã䜿ãåã«åŸ®éãªäžé§ãã¯ãããŸããã 258,259ã§ç Žç¶»ã
for(n=257,260,print(n";"floor(10^(-36)+log(10^n)/log(10))))
257;257
258;257
259;258
260;260
No.2187Dengan kesaktian Indukmu2024幎9æ20æ¥ 10:21
PARI ã«ãŠã
n = 308 ãŸã§ã®ç¯å²ã§åŸ®å°éã足ããã¹ããããŸããã
(javascriptã ãš10^308ãè¶
ãããšéäžèšç®çµæã«ç¡é倧ãçŸããæ±ãã«ãªã£ãã®ã§âŠâŠPARIã§ã¯ã©ããªã®ãããããããšããããã§ã)
埮å°éãšããŠã¯ã2 ^{-119}ãš2 ^{- 120} ãšã®ããã ã«å氎嶺ããããŸãã以äžã
? i = -120; for(n = 1, 308, if(n == floor(2^i + log(10^n)/log(10)), next, print(n, ";", floor(2^i + log(10^n)/log(10))) ) ); print("END")
ã
äžãèµ°ããããš
258;257
259;258
265;264
266;265
271;270
272;271
277;276
278;277
283;282
284;283
290;289
291;290
296;295
297;296
302;301
303;302
308;307
END
ã
ãšãªã
? i = -119; for(n = 1, 308, if(n == floor(2^i + log(10^n)/log(10)), next, print(n, ";", floor(2^i + log(10^n)/log(10))) ) ); print("END")
äžãèµ°ããããš
ã
END
ãšãªããŸãã
No.2190Dengan kesaktian Indukmu2024幎9æ22æ¥ 22:04
ããããããã®ãä»äºã§
ããã ãã®çŽ æ°ã«å¯Ÿããåžžçšå¯Ÿæ°å€ãå
±éåæ¯ã§
log2 = 360565/1197771 = 0.3010299966 (çå€ 0.3010299957)
log3 = 571482/1197771 = 0.4771212527 (çå€ 0.4771212547)
log5 = 837206/1197771 = 0.6989700034 (çå€ 0.6989700043)
log7 = 1012234/1197771 = 0.8450981031 (çå€ 0.8450980400)
log11 = 1247350/1197771 = 1.0413927203 (çå€ 1.0413926852)
log13 = 1334249/1197771 = 1.1139433164 (çå€ 1.1139433523)
log17 = 1473796/1197771 = 1.2304488922 (çå€ 1.2304489214)
log19 = 1531654/1197771 = 1.2787536182 (çå€ 1.2787536010)
log23 = 1631038/1197771 = 1.3617277426 (çå€ 1.3617278360)
log29 = 1751618/1197771 = 1.4623980711 (çå€ 1.4623979979)
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gp > abs(146/485-log(2)/log(10))
%469 = 9.3217107035117801368259509460209827810 E-7
gp > abs(73/153-log(3)/log(10))
%470 = 2.9282868735104173903973984794620415878 E-6
gp > abs(339/485-log(5)/log(10))
%471 = 9.3217107035117801368259509460209517576 E-7
gp > abs(431/510-log(7)/log(10))
%472 = 7.9857055620241233702400874249978544429 E-10
gp > abs(478/459-log(11)/log(10))
%473 = 1.6503537575300559002466218995055742675 E-6
gp > abs(743/667-log(13)/log(10))
%474 = 3.2382107964776722479812223847580763836 E-7
gp > abs(299/243-log(17)/log(10))
%475 = 3.7535188454130236161139021156448952272 E-6
gp > abs(656/513-log(19)/log(10))
%476 = 1.1643056554722575810391097558286690920 E-6
gp > abs(1103/810-log(23)/log(10))
%477 = 5.5904413551619395128281053884536869710 E-7
gp > abs(525/359-log(29)/log(10))
%478 = 2.4547234686221517882671475279717504814 E-6
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No.2179GAI2024幎9æ18æ¥ 08:34
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No.2180ãããã2024幎9æ18æ¥ 10:24
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No.2181DD++2024幎9æ18æ¥ 21:45
29ãŸã§ã®çŽ æ°ã§ãåžžçšå¯Ÿæ°ã®é£åæ°å±éãæ±ããŠã¿ãŸããããlog_{10}(7)ã®ãšãã®4813ã®ãããªå€§ããªæ°ã¯çŸããŸããã§ããããªããlog_{10}(5)=1-log_{10}(2)ãªã®ã§çç¥ããŠããŸãã
log_{10}(2)=[0;3,3,9,2,2,4,6,2,1,1,3,1,18,...]
[0;3,3,9,2,2,4,6,2,1,1,3,1]=97879/325147
=0.301029995663499893894146339963
log_{10}(3)=[0,2,10,2,2,1,13,1,7,18,...]
[0,2,10,2,2,1,13,1,7]=34367/33001
=0.477121254550546065527863343601
log_{10}(11)=[1;24,6,3,2,1,1,3,1,1,1,9,...]
[1;24,6,3,2,1,1,3,1,1,1]=22014/21139
=1.04139268507014938941244204721
log_{10}(13)=[1;1,8,1,3,2,7,1,6,16,...]
[1;1,8,1,3,2,7,1,6]=5113/4590
=1.11394335511982570806100217865
log_{10}(17)=[1,4,2,1,17,1,13,1,1,3,3,26,...]
[1;4,2,1,17,1,13,1,1,3,3]=99797/81106
=1.23045150790323773826843883313
log_{10}(19)=[1;3,1,1,2,2,1,3,2,2,1,4,1,1,1,6,1,3,1,3,1,47,...]
[1;3,1,1,2,2,1,3,2,2,1,4,1,1,1,6,1,3,1,3,1]=6497723/5081294
=1.27875360095282815755199364571
log_{10}(23)=[1;2,1,3,4,17,2,1,2,66,...]
[1;2,1,3,4,17,2,1,2]=9016/6621
=1.36172783567436943059960731007
log_{10}(29)=[[1;2,6,6,1,2,1,2,2,2,1,1,1,1,1,5,1,2,3,37,...]
[1;2,6,6,1,2,1,2,2,2,1,1,1,1,1,5,1,2,3]=5243915/3585833
=1.46239799789895402267757589380
No.2188kuiperbelt2024幎9æ22æ¥ 14:46
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log[10]2ã¯137çªç®ã5393
log[10]3ã¯562çªç®ã2788
log[10]11ã¯2179çªç®ã3864
log[10]13ã¯133çªç®ã1378
log[10]17ã¯710çªç®ã3301
log[10]19ã¯1341çªç®ã2249
log[10]23ã¯921çªç®ã2695
log[10]29ã¯352çªç®ã1901
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No.2189ãããã2024幎9æ22æ¥ 15:53