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A366075
The number of primes dividing the smallest coreful infinitary divisor of n, counted with multiplicity.
1
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
1,4
LINKS
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)).
MATHEMATICA
f[p_, e_] := 2^IntegerExponent[e, 2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(x -> 2^valuation(x, 2), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 28 2023
STATUS
approved