A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Index entries for sequences related to sigma(n)

Formula

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

A(n,k) = Sum_{j=1..n} sigma_k(j).

Example

Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
   8,  15,   37,  111,   373,   1335,  ...
  10,  21,   63,  237,   999,   4461,  ...
  14,  33,  113,  489,  2393,  12513,  ...

Mathematica

Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

Prog

(Python)
from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n, k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1, s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1, q+1) for w in range(1, s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
     return (A319649_T(k+1, n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(), 30)) # Chai Wah Wu, Oct 24 2023

Crossrefs

Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.

Cf. A082771, A109974, A319194 (diagonal).

Sequence in context: A097207 A266101 A118469 * A198553 A324288 A302917

Adjacent sequences: A319646 A319647 A319648 * A319650 A319651 A319652

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Dec 09 2018

Status

approved