This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A238442 #90 Aug 23 2024 23:23:48
%S A238442 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,
%T A238442 -6,-4,18,13,-12,-7,12,18,-8,-6,12,28,12,-15,-12,1,14,28,-13,-8,3,24,
%U A238442 14,-18,-15,4,15,24,24,-12,-13,7,1,31,24,-28,-18,6,3
%N A238442 Triangle read by rows demonstrating Euler's pentagonal theorem for the sum of divisors.
%C A238442 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.
%C A238442 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).
%C A238442 The first element of column k is A057077(k-1)*A001318(k)which is also the last term of row A001318(k).
%C A238442 For Euler's pentagonal theorem for the partition numbers see A175003.
%C A238442 Note that both of Euler's pentagonal theorems refer to generalized pentagonal numbers (A001318), not to pentagonal numbers (A000326).
%H A238442 L. Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
%H A238442 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>
%H A238442 L. Euler, <a href="http://eulerarchive.maa.org//pages/E175.html">Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs</a>
%H A238442 L. Euler, <a href="http://eulerarchive.maa.org/docs/translations/E175en.pdf">Discovery of a most extraordinary law of numbers, relating to the sum of their divisors</a>
%H A238442 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a>, p. 8.
%H A238442 L. Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H A238442 L. Euler, J. Bell, <a href="http://arxiv.org/abs/math/0507201">A demonstration of a theorem on the order observed in the sums of divisors</a>, arXiv:math/0507201 [math.HO], 2005-2009.
%H A238442 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A238442 T(n,k) = A057077(k-1)*A001318(k), if n = A001318(k) and k = A235963(n). Otherwise T(n,k) = A057077(k-1)*A000203(n - A001318(k)), n >= 1, 1 <= k <= A235963(n).
%e A238442 Triangle begins:
%e A238442 1;
%e A238442 1, 2;
%e A238442 3, 1;
%e A238442 4, 3;
%e A238442 7, 4, -5;
%e A238442 6, 7, -1;
%e A238442 12, 6, -3, -7;
%e A238442 8, 12, -4, -1;
%e A238442 15, 8, -7, -3;
%e A238442 13, 15, -6, -4;
%e A238442 18, 13, -12, -7;
%e A238442 12, 18, -8, -6, 12;
%e A238442 28, 12, -15, -12, 1;
%e A238442 14, 28, -13, -8, 3;
%e A238442 24, 14, -18, -15, 4, 15;
%e A238442 24, 24, -12, -13, 7, 1;
%e A238442 31, 24, -28, -18, 6, 3;
%e A238442 18, 31, -14, -12, 12, 4;
%e A238442 39, 18, -24, -28, 8, 7;
%e A238442 20, 39, -24, -14, 15, 6;
%e A238442 42, 20, -31, -24, 13, 12;
%e A238442 32, 42, -18, -24, 18, 8, -22;
%e A238442 36, 32, -39, -31, 12, 15, -1;
%e A238442 24, 36, -20, -18, 28, 13, -3;
%e A238442 60, 24, -42, -39, 14, 18, -4;
%e A238442 31, 60, -32, -20, 24, 12, -7, -26;
%e A238442 ...
%e A238442 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.
%e A238442 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.
%t A238442 rows = m = 18;
%t A238442 a057077[n_] := {1, 1, -1, -1}[[Mod[n, 4] + 1]];
%t A238442 a001318[n_] := (1/8)((2n + 1) Mod[n, 2] + 3n^2 + 2n);
%t A238442 a235963[n_] := Flatten[Table[k, {k, 0, m}, {(k+1)/(Mod[k, 2]+1)}]][[n+1]];
%t A238442 T[n_, k_] := If[n == a001318[k] && k == a235963[n], a001318[k] a057077[k - 1], a057077[k - 1] DivisorSigma[1, n - a001318[k]]];
%t A238442 Table[T[n, k], {n, 1, m}, {k, 1, a235963[n]}] // Flatten (* _Jean-François Alcover_, Nov 29 2018 *)
%o A238442 (GW-BASIC)
%o A238442 10 'A program with four A-numbers. _Omar E. Pol_, Feb 26 2014
%o A238442 20 Dim A000203(30), A001318(10), A057077(30), A235963(30), T(30,10)
%o A238442 30 For n = 1 to 26
%o A238442 40 For k = 1 to A235963(n)
%o A238442 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))
%o A238442 60 print T(n,k);
%o A238442 70 next k
%o A238442 80 print
%o A238442 90 next n
%o A238442 100 End
%Y A238442 Row sums give A000203, the sum of divisors of n.
%Y A238442 Row n has length A235963(n).
%Y A238442 Cf. A001318, A027750, A057077, A175003, A196020, A237270, A237273.
%K A238442 sign,tabf
%O A238442 1,3
%A A238442 _Omar E. Pol_, Feb 26 2014