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A360330 a(n) is the number of divisors of n that have only prime factors that are not prime-indexed primes.

%I A360330 #17 Feb 04 2023 14:14:30
%S A360330 1,2,1,3,1,2,2,4,1,2,1,3,2,4,1,5,1,2,2,3,2,2,2,4,1,4,1,6,2,2,1,6,1,2,
%T A360330 2,3,2,4,2,4,1,4,2,3,1,4,2,5,3,2,1,6,2,2,1,8,2,4,1,3,2,2,2,7,2,2,1,3,
%U A360330 2,4,2,4,2,4,1,6,2,4,2,5,1,2,1,6,1,4,2
%N A360330 a(n) is the number of divisors of n that have only prime factors that are not prime-indexed primes.
%C A360330 Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime factors that are not prime-indexed primes.
%H A360330 Amiram Eldar, <a href="/A360330/b360330.txt">Table of n, a(n) for n = 1..10000</a>
%F A360330 a(n) = 1 if and only if n is in A076610.
%F A360330 a(n) = A000005(n) if and only if n is in A320628.
%F A360330 a(n) = A000005(A360329(n)).
%F A360330 Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and e+1 otherwise (A007821).
%p A360330 a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1, i[2]+1), i=ifactors(n)[2]):
%p A360330 seq(a(n), n=1..87);  # _Alois P. Heinz_, Feb 03 2023
%t A360330 f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, e+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360330 (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, e[i]+1));}
%Y A360330 Cf. A000005, A006450, A007821, A076610, A320628, A360329, A360331.
%K A360330 nonn,mult
%O A360330 1,2
%A A360330 _Amiram Eldar_, Feb 03 2023