A334393 -id:A334393 - cp's OEIS frontend

A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A122810 at n = 48, and from A318322 at n = 64.

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

Cf. A000720, A001221, A005117, A077761, A334393, A366988, A366991, A366992, A366994.

Cf. A122810, A318322.

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

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

Additive with a(p^e) = A000720(e) + 1.
a(n) = 1 is and only if n is squarefree (A005117) > 1.
a(n) = A366988(n) + A001221(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} P(p) = 0.67167522222173297323..., and P(s) is the prime zeta function.

A334395 Partial products of A334393.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 6720, 60480, 665280, 8648640, 147026880, 2793510720, 64250746560, 1606268664000, 43369253928000, 1257708363912000, 38988959281272000, 1247646697000704000, 46162927789026048000, 1892680039350067968000, 81385241692052922624000

Offset: 1

Kevin Foote, Apr 26 2020

nonn

			a(6) = 1*2*3*4*5*7 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..358

Cf. A008578, A024923, A334393.

Rest @ FoldList[Times, 1, Select[Range[43], Length[(f = FactorInteger[#])] == 1 && ((e = f[[1, 2]]) == 1 || PrimeQ[e]) &]] (* Amiram Eldar, May 11 2020 *)

isok(n) = if (n==1, return (1)); my(k=isprimepower(n)); (k==1) || isprime(k); \\ A334393
lista(nn) = {my(v = select(x->isok(x), [1..nn]), p=1); for (n=1, #v, p *= v[n]; print1(p, ", "););} \\ Michel Marcus, May 11 2020

a(n) = Product_{i=1..n} A334393(i).

cp's OEIS Frontend

A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

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