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

1,2

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

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).

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]

Prog

(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))); }

Crossrefs

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

Sequence in context: A065745 A361438 A268670 * A227873 A117677 A324377

Adjacent sequences: A360328 A360329 A360330 * A360332 A360333 A360334

Keyword

nonn,mult

Author

Amiram Eldar, Feb 03 2023

Status

approved