OFFSET
1,3
COMMENTS
The law found by Leonhard Euler for the sum of divisors of n is that S(n) = S(n - 1) + S(n - 2) - S(n - 5) - S(n - 7) + S(n - 12) + S(n - 15) - S(n - 22) - S(n - 26) + S(n - 35) + S(n - 40) + ..., where the constants are the positive generalized pentagonal numbers, and S(0) = n, which is also a positive member of A001318.
Therefore column k lists A001318(k) together with the elements of A000203, starting at row A001318(k), but with all elements of column k multiplied by A057077(k-1).
The first element of column k is A057077(k-1)*A001318(k)which is also the last term of row A001318(k).
For Euler's pentagonal theorem for the partition numbers see A175003.
LINKS
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
L. Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs
L. Euler, Observatio de summis divisorum, p. 8.
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
L. Euler, J. Bell, A demonstration of a theorem on the order observed in the sums of divisors, arXiv:math/0507201 [math.HO], 2005-2009.
FORMULA
EXAMPLE
Triangle begins:
1;
1, 2;
3, 1;
4, 3;
7, 4, -5;
6, 7, -1;
12, 6, -3, -7;
8, 12, -4, -1;
15, 8, -7, -3;
13, 15, -6, -4;
18, 13, -12, -7;
12, 18, -8, -6, 12;
28, 12, -15, -12, 1;
14, 28, -13, -8, 3;
24, 14, -18, -15, 4, 15;
24, 24, -12, -13, 7, 1;
31, 24, -28, -18, 6, 3;
18, 31, -14, -12, 12, 4;
39, 18, -24, -28, 8, 7;
20, 39, -24, -14, 15, 6;
42, 20, -31, -24, 13, 12;
32, 42, -18, -24, 18, 8, -22;
36, 32, -39, -31, 12, 15, -1;
24, 36, -20, -18, 28, 13, -3;
60, 24, -42, -39, 14, 18, -4;
31, 60, -32, -20, 24, 12, -7, -26;
...
For n = 21 the sum of divisors of 21 is 1 + 3 + 7 + 21 = 32. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 21 is S_21 = S_20 + S_19 - S_16 - S_14 + S_9 + S_6, the same as the sum of the 21st row of triangle: 42 + 20 - 31 - 24 + 13 + 12 = 32, equaling the sum of divisors of 21.
For n = 22 the sum of divisors of 22 is 1 + 2 + 11 + 22 = 36. On the other hand, from Euler's Pentagonal Number Theorem we have that the sum of divisors of 22 is S_22 = S_21 + S_20 - S_17 - S_15 + S_10 + S_7 - S_0, the same as the sum of the 22nd row of triangle is 32 + 42 - 18 - 24 + 18 + 8 - 22 = 36, equaling the sum of divisors of 22. Note that S_0 = n, hence in this case S_0 = 22.
MATHEMATICA
rows = m = 18;
a057077[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]];
a001318[n_] := (1/8)((2n + 1) Mod[n, 2] + 3n^2 + 2n);
a235963[n_] := Flatten[Table[k, {k, 0, m}, {(k+1)/(Mod[k, 2]+1)}]][[n+1]];
T[n_, k_] := If[n == a001318[k] && k == a235963[n], a001318[k] a057077[k - 1], a057077[k - 1] DivisorSigma[1, n - a001318[k]]];
Table[T[n, k], {n, 1, m}, {k, 1, a235963[n]}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)
PROG
(GW-BASIC)
10 'A program with four A-numbers. Omar E. Pol, Feb 26 2014
30 For n = 1 to 26
40 For k = 1 to A235963(n)
50 If n = A001318(k) and k = A235963(n) then T(n, k) = A057077(k-1)*A001318(k) else T(n, k) = A057077(k-1)*A000203(n - A001318(k))
60 print T(n, k);
70 next k
80 print
90 next n
100 End
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Omar E. Pol, Feb 26 2014
STATUS
approved