A360331 - OEIS

%I #14 Feb 04 2023 14:14:26

%S 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 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 20,90,1,7,62,3,8,127,14,3,1,7,24,24,72,15,74,114,1

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

%C 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 Amiram Eldar, <a href="/A360331/b360331.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 1 if and only if n is in A076610.

%F a(n) = A000203(n) if and only if n is in A320628.

%F a(n) = A000203(A360329(n)).

%F Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and (p^(e+1)-1)/(p-1) otherwise (A007821).

%p a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1,

%p    (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2]):

%p seq(a(n), n=1..75);  # _Alois P. Heinz_, Feb 03 2023

%t 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 (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 Cf. A000203, A006450, A007821, A076610, A320628, A360329, A360330.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, Feb 03 2023