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A372503
The number of prime powers that are noninfinitary divisors of n.
1
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0
OFFSET
1,16
COMMENTS
First differs from A318499 at n = 32.
LINKS
FORMULA
Additive with a(p^e) = e - 2^A000120(e) + 1 = A048967(e).
a(n) = A001222(n) - A349258(n).
a(n) = 0 if and only if n is in A036537.
a(n) > 0 if and only if n is in A162643.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.4917971717413486467..., where f(x) = 1/(1-x) - (1-x) * Product_{k>=0} (1 + 2*x^(2^k)).
MATHEMATICA
f[p_, e_] := e + 1 - 2^DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(x -> x + 1 - 1 << hammingweight(x), factor(n)[, 2]));
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, May 04 2024
STATUS
approved