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

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, 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, 73, 10, 8, 1, 0, 1, 1, 257, 1, 794, 1, 273, 28, 30, 1, 5

Offset

1,10

Comments

A(n,k) is the sum of k-th powers of proper divisors of n.

Links

Table of n, a(n) for n=1..78.

Eric Weisstein's World of Mathematics, Proper divisors

Index entries for sequences related to sums of divisors

Formula

G.f. of column k: Sum_{j>=1} j^k*x^(2*j)/(1 - x^j).

Dirichlet g.f. of column k: zeta(s-k)*(zeta(s) - 1).

A(n,k) = 1 if n is prime.

Example

Square array begins:
  0,  0,   0,   0,   0,    0,  ...
  1,  1,   1,   1,   1,    1,  ...
  1,  1,   1,   1,   1,    1,  ...
  2,  3,   5,   9,  17,   33,  ...
  1,  1,   1,   1,   1,    1,  ...
  3,  6,  14,  36,  98,  276,  ...

Mathematica

Table[Function[k, DivisorSigma[k, n] - n^k][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
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

Crossrefs

Columns k=0..5 give A032741, A001065, A067558, A276634, A279363, A279364.

Cf. A109974, A285425, A286880, A321259 (diagonal).

Sequence in context: A217760 A339218 A263412 * A331510 A319854 A124035

Adjacent sequences: A321255 A321256 A321257 * A321259 A321260 A321261

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Nov 01 2018

Status

approved