A366075 - OEIS

%I #12 Sep 29 2023 04:09:51

%S 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,

%T 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,

%U 2,3,1,3,1,2,3,3,2,3,1,5,4,2,1,4,2,2,2

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

%H Amiram Eldar, <a href="/A366075/b366075.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F 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)).

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

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

%Y Cf. A001222, A006519, A246551, A365296.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Sep 28 2023