A366075 The number of primes dividing the smallest coreful infinitary divisor of n, counted with multiplicity.

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

Offset: 1

Author

Amiram Eldar, Sep 28 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A001222, A006519, A246551, A365296.

Programs

Mathematica

f[p_, e_] := 2^IntegerExponent[e, 2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecsum(apply(x -> 2^valuation(x, 2), factor(n)[, 2]));

Formula

a(n) = A001222(A365296(n)).

Additive with a(p^e) = A006519(e).

a(n) = 1 if and only if n is in A246551.

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.42540262231508387576..., where f(x) = -x + (1-x) * Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)).