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A353551 a(n) = Sum_{k=1..n} tau(k^3), where tau is the number of divisors function A000005.

%I A353551 #24 May 10 2022 13:48:21
%S A353551 0,1,5,9,16,20,36,40,50,57,73,77,105,109,125,141,154,158,186,190,218,
%T A353551 234,250,254,294,301,317,327,355,359,423,427,443,459,475,491,540,544,
%U A353551 560,576,616,620,684,688,716,744,760,764,816,823,851,867,895,899,939,955,995
%N A353551 a(n) = Sum_{k=1..n} tau(k^3), where tau is the number of divisors function A000005.
%H A353551 Karl-Heinz Hofmann, <a href="/A353551/b353551.txt">Table of n, a(n) for n = 0..10000</a>
%F A353551 a(n) = Sum_{k=1..n} tau(k^3).
%F A353551 a(n) = a(n-1) + A048785(n) for n >= 1, a(0) = 0.
%e A353551   A048785(0) = 0
%e A353551 + A048785(1) = 1
%e A353551 + A048785(2) = 4
%e A353551 + A048785(3) = 4
%e A353551 ------------------
%e A353551 = A353551(3) = 9
%p A353551 a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+numtheory[tau](n^3)) end:
%p A353551 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 08 2022
%t A353551 Accumulate[Join[{0}, Table[DivisorSigma[0, k^3], {k, 1, 50}]]] (* _Amiram Eldar_, May 08 2022 *)
%o A353551 (Python) from sympy import divisor_count
%o A353551 def A048785(n): return divisor_count(n**3)
%o A353551 def A353551(n): return sum(A048785(n) for n in range(1, n))
%o A353551 print([A353551(n) for n in range(1, 58)])
%o A353551 (PARI) a(n) = sum(k=1, n, numdiv(k^3)); \\ _Michel Marcus_, May 08 2022
%o A353551 (Python)
%o A353551 from math import prod
%o A353551 from sympy import factorint
%o A353551 def A353551(n): return sum(prod(3*e+1 for e in factorint(k).values()) for k in range(1,n+1)) # _Chai Wah Wu_, May 10 2022
%Y A353551 Partial sums of A048785.
%Y A353551 Cf. A000005, A006218, A061503 (squares).
%K A353551 nonn
%O A353551 0,3
%A A353551 _Karl-Heinz Hofmann_, May 07 2022