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A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).

%I A321348 #28 Dec 14 2022 09:08:31
%S A321348 1,4,5,15,7,64,9,52,30,144,13,546,15,256,289,165,19,1140,21,1386,529,
%T A321348 576,25,3848,78,784,166,2610,31,32768,33,486,1225,1296,1369,12321,39,
%U A321348 1600,1681,10248,43,85184,45,6210,6486,2304,49,24250,150,7956
%N A321348 a(n) = Sum_{d|n} tau(d^n), where tau() is the number of divisors (A000005).
%C A321348 a(n) is prime iff n is in A001359, which makes the sequence a supersequence of A006512. - _Ivan N. Ianakiev_, Nov 07 2018
%H A321348 Vincenzo Librandi, <a href="/A321348/b321348.txt">Table of n, a(n) for n = 1..5000</a>
%F A321348 a(n) = [x^n] Sum_{k>=1} tau(k^n)*x^k/(1 - x^k).
%F A321348 If n = Product (p_j^k_j) then a(n) = Product ((k_j + 1)*(n*k_j + 2)/2).
%F A321348 a(prime(n)) = prime(n) + 2 = A052147(n). - _Michel Marcus_, Nov 25 2018
%p A321348 with(numtheory): seq(coeff(series(add(tau(k^n)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 50); # _Muniru A Asiru_, Nov 25 2018
%t A321348 Table[Sum[DivisorSigma[0, d^n], {d, Divisors[n]}], {n, 50}]
%t A321348 a[n_] := Times @@ ((#[[2]] + 1) (n #[[2]] + 2)/2 & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 50}]
%o A321348 (PARI) a(n) = sumdiv(n, d, numdiv(d^n)); \\ _Michel Marcus_, Nov 06 2018
%o A321348 (Magma) [&+[NumberOfDivisors(d^n): d in Divisors(n)]: n in [1..50]]; // _Vincenzo Librandi_, Nov 08 2018
%o A321348 (Python)
%o A321348 from math import prod
%o A321348 from sympy import factorint
%o A321348 def A321348(n): return prod((e+1)*(n*e+2)>>1 for e in factorint(n).values()) # _Chai Wah Wu_, Dec 13 2022
%Y A321348 Cf. A000005, A007425, A035116, A061391.
%Y A321348 Cf. A001359, A006512, A052147.
%K A321348 nonn
%O A321348 1,2
%A A321348 _Ilya Gutkovskiy_, Nov 06 2018