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A368544 The number of divisors of n whose prime factors are all of the form k^2+1.

%I A368544 #11 Dec 30 2023 17:22:30
%S A368544 1,2,1,3,2,2,1,4,1,4,1,3,1,2,2,5,2,2,1,6,1,2,1,4,3,2,1,3,1,4,1,6,1,4,
%T A368544 2,3,2,2,1,8,1,2,1,3,2,2,1,5,1,6,2,3,1,2,2,4,1,2,1,6,1,2,1,7,2,2,1,6,
%U A368544 1,4,1,4,1,4,3,3,1,2,1,10,1,2,1,3,4,2,1
%N A368544 The number of divisors of n whose prime factors are all of the form k^2+1.
%C A368544 The number of terms of A180252 that divide n.
%H A368544 Amiram Eldar, <a href="/A368544/b368544.txt">Table of n, a(n) for n = 1..10000</a>
%F A368544 Multiplicative with a(p^e) = e+1 if p is of the form k^2+1, and 1 otherwise.
%F A368544 a(n) >= 1, with equality if and only if all the prime factors of n are in A070303.
%F A368544 a(n) <= A000005(n), with equality if and only if n is in A180252.
%F A368544 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{k in A005574} (1 + 1/k^2) = 2.80986546... .
%t A368544 q[n_] := AllTrue[FactorInteger[n][[;; , 1]], IntegerQ[Sqrt[# - 1]] &]; f[p_, e_] := If[q[p], e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368544 (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(issquare(f[i,1]-1), f[i,2] + 1, 1))};
%o A368544 (Python)
%o A368544 from math import prod
%o A368544 from sympy import factorint
%o A368544 from sympy.ntheory.primetest import is_square
%o A368544 def A368544(n): return prod(e+1 for p, e in factorint(n).items() if is_square(p-1)) # _Chai Wah Wu_, Dec 30 2023
%Y A368544 Cf. A002496, A005574, A070303, A180252.
%K A368544 nonn,easy,mult
%O A368544 1,2
%A A368544 _Amiram Eldar_, Dec 29 2023