A083064 -id:A083064 - cp's OEIS frontend

A085388 First differences of n^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 8, 0, 1, 5, 20, 48, 54, 16, 0, 1, 6, 30, 100, 192, 162, 32, 0, 1, 7, 42, 180, 500, 768, 486, 64, 0, 1, 8, 56, 294, 1080, 2500, 3072, 1458, 128, 0, 1, 9, 72, 448, 2058, 6480, 12500, 12288, 4374, 256, 0, 1, 10, 90, 648

Offset: 1

Paul Barry, Jun 30 2003

T(n,k) is the number of k-digit numbers in base n; n,k >= 2. - Mohammed Yaseen, Nov 11 2022

			Rows begin
  1,   0,   0,   0,   0, ...
  1,   1,   2,   4,   8, ...
  1,   2,   6,  18,  54, ...
  1,   3,  12,  48, 192, ...
  1,   4,  20, 100, 500, ...

Rows include A011782, A025192, A002001, A005054, A052934, A055272, A055274, A055275, A052268, A055276.
Diagonals include A053506, A085389, A085390.
Row-wise binomial transform is A083064.

T(n,k) = (n-1)*n^(k-1) + 0^k/n. - Corrected by Mohammed Yaseen, Nov 11 2022

T(n,0) = 1; T(n,k) = n^k - n^(k-1) for k >= 1. - Mohammed Yaseen, Nov 11 2022

Offset corrected by Mohammed Yaseen, Nov 11 2022

A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

Original entry on oeis.org

1, 1, 5, 1, 19, -19, 41, 1, 41, -17, 109, -115, 155, -7, 47, 1, 271, -199, 341, -161, 141, 37, 505, -499, 469, 71, 365, -199, 811, -1021, 929, 1, 449, 163, 683, -1159, 1331, 221, 663, -737, 1639, -1659, 1805, -251, 299, 361, 2161, -2035, 2001, -467, 1211, -265, 2755, -1819, 1927, -967, 1545, 631, 3421, -5293, 3659, 737

Offset: 1

Antti Karttunen, Nov 23 2024

sign

It is conjectured that 1's occur only when n is two's power (A000079), and that there are no -1's in this sequence. See comments in A033879 and in A337339.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000290, A000079 (conjectured to give positions of all 1's), A033879, A378231 [= a(A003961(n))].
Cf. A000012, A083884.
Cf. also square array A083064.

Table[2n^2-DivisorSigma[1, n^2], {n, 62}] (* James C. McMahon, Nov 24 2024 *)

A033879(n) = (n+n-sigma(n));
A377879(n) = A033879(n*n);

a(n) = A033879(A000290(n)).

cp's OEIS Frontend

A085388 First differences of n^k.

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A377879 Deficiency of squares: a(n) = 2n^2 - sigma(n^2).

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