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A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.

%I A321258 #6 Feb 16 2025 08:33:57
%S A321258 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 A321258 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 A321258 73,10,8,1,0,1,1,257,1,794,1,273,28,30,1,5
%N A321258 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = sigma_k(n) - n^k.
%C A321258 A(n,k) is the sum of k-th powers of proper divisors of n.
%H A321258 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProperDivisor.html">Proper divisors</a>
%H A321258 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A321258 G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).
%F A321258 Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).
%F A321258 A(n,k) = 1 if n is prime.
%e A321258 Square array begins:
%e A321258   0,  0,   0,   0,   0,    0,  ...
%e A321258   1,  1,   1,   1,   1,    1,  ...
%e A321258   1,  1,   1,   1,   1,    1,  ...
%e A321258   2,  3,   5,   9,  17,   33,  ...
%e A321258   1,  1,   1,   1,   1,    1,  ...
%e A321258   3,  6,  14,  36,  98,  276,  ...
%t A321258 Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
%t A321258 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 A321258 Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.
%Y A321258 Cf. A109974, A285425, A286880, A321259 (diagonal).
%K A321258 nonn,tabl
%O A321258 1,10
%A A321258 _Ilya Gutkovskiy_, Nov 01 2018