OFFSET
1,2
COMMENTS
Gives an identity for sigma(n). Alternating sum of row n equals A000203(n), the sum of the divisors of n.
Column 1 is A000716, but here the offset is 1 not 0.
The 1st element of column k is A000330(k).
The 2nd element of column k is A059270(k).
The 3rd element of column k is A220443(k).
The partial sums of column k give the k-th column of A249120.
This triangle has been constructed after Mircea Merca's formula for A000203.
From Omar E. Pol, May 05 2022: (Start)
In the Honda-Yoda paper see "3. String theory and Riemann hypothesis". The coefficients that are mentioned in 3.11 are the first 16 terms of A000716, the coefficients that are mentioned in 3.12 are the first 5 terms of A000330, and the coefficients that are mentioned in 3.13 are the first 16 terms of A000203.
LINKS
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], (2022), pp. 5-6.
EXAMPLE
Triangle begins:
1;
3;
9, 5;
22, 15;
51, 45;
108, 110, 14;
221, 255, 42;
429, 540, 126;
810, 1105, 308;
1479, 2145, 714, 30;
2640, 4050, 1512, 90;
4599, 7395, 3094, 270;
7868, 13200, 6006, 660;
13209, 22995, 11340, 1530;
21843, 39340, 20706, 3240, 55;
35581, 66045, 36960, 6630, 165;
57222, 109215, 64386, 12870, 495;
90882, 177905, 110152, 24300, 1210;
142769, 286110, 184926, 44370, 2805;
221910, 454410, 305802, 79200, 5940;
341649, 713845, 498134, 137970, 12155, 91;
...
For n = 6 the divisors of 6 are 1, 2, 3, 6, so the sum of the divisors of 6 is 1 + 2 + 3 + 6 = 12. On the other hand, the 6th row of the triangle is 108, 110, 14, so the alternating row sum is 108 - 110 + 14 = 12, equaling the sum of the divisors of 6.
For n = 15 the divisors of 15 are 1, 3, 5, 15, so the sum of the divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand, the 15th row of the triangle is 21843, 39340, 20706, 3240, 55, so the alternating row sum is 21843 - 39340 + 20706 - 3240 + 55 = 24, equaling the sum of the divisors of 15.
PROG
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 14 2014
STATUS
approved