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

%I A343657 #11 Jun 04 2021 18:07:09
%S A343657 1,2,4,7,12,18,27,39,56,77,103,134,174,223,283,356,445,547,666,802,
%T A343657 959,1139,1344,1574,1835,2128,2454,2815,3213,3648,4126,4653,5239,5888,
%U A343657 6608,7407,8298,9288,10385,11597,12936,14408,16025,17799,19746,21882,24221
%N A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.
%H A343657 Chai Wah Wu, <a href="/A343657/b343657.txt">Table of n, a(n) for n = 1..10000</a>
%F A343657 a(n) = Sum_{k=1..n} A000005(k^(n-k)).
%e A343657 The a(7) = 27 divisors:
%e A343657   1  32  81  64  25  6  1
%e A343657      16  27  32  5   3
%e A343657      8   9   16  1   2
%e A343657      4   3   8       1
%e A343657      2   1   4
%e A343657      1       2
%e A343657              1
%t A343657 Total/@Table[DivisorSigma[0,k^(n-k)],{n,30},{k,n}]
%o A343657 (Python 3.8+)
%o A343657 from math import prod
%o A343657 from sympy import factorint
%o A343657 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
%Y A343657 Antidiagonal row sums (row sums of the triangle) of A343656.
%Y A343657 Dominated by A343661.
%Y A343657 A000005(n) counts divisors of n.
%Y A343657 A000312(n) = n^n.
%Y A343657 A007318(n,k) counts k-sets of elements of {1..n}.
%Y A343657 A009998(n,k) = n^k (as an array, offset 1).
%Y A343657 A059481(n,k) counts k-multisets of elements of {1..n}.
%Y A343657 A343658(n,k) counts k-multisets of divisors of n.
%Y A343657 Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.
%K A343657 nonn
%O A343657 1,2
%A A343657 _Gus Wiseman_, Apr 29 2021