%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