A360327 a(n) is the sum of divisors of n that have only prime-indexed prime factors.

1, 1, 4, 1, 6, 4, 1, 1, 13, 6, 12, 4, 1, 1, 24, 1, 18, 13, 1, 6, 4, 12, 1, 4, 31, 1, 40, 1, 1, 24, 32, 1, 48, 18, 6, 13, 1, 1, 4, 6, 42, 4, 1, 12, 78, 1, 1, 4, 1, 31, 72, 1, 1, 40, 72, 1, 4, 1, 60, 24, 1, 32, 13, 1, 6, 48, 68, 18, 4, 6, 1, 13, 1, 1, 124, 1, 12

Offset: 1

Author

Amiram Eldar, Feb 03 2023

Keywords

• nonn
• mult

Comments

Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime-indexed prime factors.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A006450, A076610, A320628, A360325, A360326.

Programs

Mathematica

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}

Formula

a(n) = 1 if and only if n is in A320628.

a(n) = A000203(n) if and only if n is in A076610.

a(n) = A000203(A360325(n)).

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