A360330 a(n) is the number of divisors of n that have only prime factors that are not prime-indexed primes.

1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 2, 2, 2, 4, 1, 4, 1, 6, 2, 2, 1, 6, 1, 2, 2, 3, 2, 4, 2, 4, 1, 4, 2, 3, 1, 4, 2, 5, 3, 2, 1, 6, 2, 2, 1, 8, 2, 4, 1, 3, 2, 2, 2, 7, 2, 2, 1, 3, 2, 4, 2, 4, 2, 4, 1, 6, 2, 4, 2, 5, 1, 2, 1, 6, 1, 4, 2

Offset

1,2

Comments

Equivalently, a(n) is the number 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) = A000005(n) if and only if n is in A320628.

a(n) = A000005(A360329(n)).

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

Maple

a:= n-> mul(`if`(isprime(numtheory[pi](i[1])), 1, i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..87);  # Alois P. Heinz, Feb 03 2023

Mathematica

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, e+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, e[i]+1)); }

Crossrefs

Cf. A000005, A006450, A007821, A076610, A320628, A360329, A360331.

Sequence in context: A035942 A329622 A036989 * A035197 A227872 A323165

Adjacent sequences: A360327 A360328 A360329 * A360331 A360332 A360333

Keyword

nonn,mult

Author

Amiram Eldar, Feb 03 2023

Status

approved