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A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.

%I A360331 #14 Feb 04 2023 14:14:26
%S A360331 1,3,1,7,1,3,8,15,1,3,1,7,14,24,1,31,1,3,20,7,8,3,24,15,1,42,1,56,30,
%T A360331 3,1,63,1,3,8,7,38,60,14,15,1,24,44,7,1,72,48,31,57,3,1,98,54,3,1,120,
%U A360331 20,90,1,7,62,3,8,127,14,3,1,7,24,24,72,15,74,114,1
%N A360331 a(n) is the sum of divisors of n that have only prime factors that are not prime-indexed primes.
%C A360331 Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime factors that are not prime-indexed primes.
%H A360331 Amiram Eldar, <a href="/A360331/b360331.txt">Table of n, a(n) for n = 1..10000</a>
%F A360331 a(n) = 1 if and only if n is in A076610.
%F A360331 a(n) = A000203(n) if and only if n is in A320628.
%F A360331 a(n) = A000203(A360329(n)).
%F A360331 Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and (p^(e+1)-1)/(p-1) otherwise (A007821).
%p A360331 a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1,
%p A360331    (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2]):
%p A360331 seq(a(n), n=1..75);  # _Alois P. Heinz_, Feb 03 2023
%t A360331 f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360331 (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1)));}
%Y A360331 Cf. A000203, A006450, A007821, A076610, A320628, A360329, A360330.
%K A360331 nonn,mult
%O A360331 1,2
%A A360331 _Amiram Eldar_, Feb 03 2023