A366990 -id:A366990 - cp's OEIS frontend

A366988 The number of prime powers of prime numbers (A053810) that divide n.

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Oct 31 2023

a(n) depends only on the prime signature of n.

Every nonnegative number appears in the sequence of record values. k >= 1 first occurs at n = 2^prime(k) (A034785).

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

Cf. A000720, A005117, A034785, A053810, A095691, A366989, A366990, A366993.

f[p_, e_] := PrimePi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i, 2]));}

Additive with a(p^e) = A000720(e).
a(n) = 0 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} P(p) = 0.67167522222173297323..., where P(s) is the prime zeta function.

A366993 The largest divisor of n that is a term of A056166.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 8, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 32, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 8, 27, 1, 1, 4

Offset: 1

Amiram Eldar, Oct 31 2023

The number of these divisors is A095691(n) and their sum is A366990(n).

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

Cf. A005117, A007917, A056166, A095691, A366988, A366990.

f[p_, e_] := p^If[e == 1, 0, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^precprime(f[i, 2]));}

Multiplicative with a(p) = 1 and a(p^e) = p^A007917(e).
a(n) <= n, with equality if and only if n is in A056166.
a(n) >= 1, with equality if and only if n is squarefree (A005117).

cp's OEIS Frontend

A366988 The number of prime powers of prime numbers (A053810) that divide n.

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