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%I #5 Nov 01 2018 18:21:03
%S 0,0,1,0,1,1,0,1,1,2,0,1,1,3,1,0,1,1,5,1,3,0,1,1,9,1,6,1,0,1,1,17,1,
%T 14,1,3,0,1,1,33,1,36,1,7,2,0,1,1,65,1,98,1,21,4,3,0,1,1,129,1,276,1,
%U 73,10,8,1,0,1,1,257,1,794,1,273,28,30,1,5
%N Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.
%C A(n,k) is the sum of k-th powers of proper divisors of n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProperDivisor.html">Proper divisors</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
%F Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
%F A(n,k) = 1 if n is prime.
%e Square array begins:
%e 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 2, 3, 5, 9, 17, 33, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 3, 6, 14, 36, 98, 276, ...
%t Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%t Table[Function[k, SeriesCoefficient[Sum[j^k x^(2 j)/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%Y Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.
%Y Cf. A109974, A285425, A286880, A321259 (diagonal).
%K nonn,tabl
%O 1,10
%A _Ilya Gutkovskiy_, Nov 01 2018
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