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

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

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 factors that are not prime-indexed primes.

Links

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

Crossrefs

Cf. A000203, A006450, A007821, A076610, A320628, A360329, A360330.

Programs

Maple

a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1,
   (i[1]^(i[2]+1)-1)/(i[1]-1)), i=ifactors(n)[2]):
seq(a(n), n=1..75);  # Alois P. Heinz, Feb 03 2023

Mathematica

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]

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)));}

Formula

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

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

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

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