A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

1, 2, 4, 7, 12, 18, 27, 39, 56, 77, 103, 134, 174, 223, 283, 356, 445, 547, 666, 802, 959, 1139, 1344, 1574, 1835, 2128, 2454, 2815, 3213, 3648, 4126, 4653, 5239, 5888, 6608, 7407, 8298, 9288, 10385, 11597, 12936, 14408, 16025, 17799, 19746, 21882, 24221

Offset

1,2

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{k=1..n} A000005(k^(n-k)).

Example

The a(7) = 27 divisors:
  1  32  81  64  25  6  1
     16  27  32  5   3
     8   9   16  1   2
     4   3   8       1
     2   1   4
     1       2
             1

Mathematica

Total/@Table[DivisorSigma[0, k^(n-k)], {n, 30}, {k, n}]

Prog

(Python 3.8+)
from math import prod
from sympy import factorint
def A343657(n): return 1 if n == 1 else 2 + sum((prod(d*(n-k)+1 for d in factorint(k).values())) for k in range(2, n)) # Chai Wah Wu, Jun 03 2021

Crossrefs

Antidiagonal row sums (row sums of the triangle) of A343656.

Dominated by A343661.

A000005(n) counts divisors of n.

A000312(n) = n^n.

A007318(n,k) counts k-sets of elements of {1..n}.

A009998(n,k) = n^k (as an array, offset 1).

A059481(n,k) counts k-multisets of elements of {1..n}.

A343658(n,k) counts k-multisets of divisors of n.

Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.

Sequence in context: A049703 A175812 A002621 * A363211 A033500 A003318

Adjacent sequences: A343654 A343655 A343656 * A343658 A343659 A343660

Keyword

nonn

Author

Gus Wiseman, Apr 29 2021

Status

approved