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A366074
The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.
5
0, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 3, 1, 0, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 3, 1, 2, 2, 0, 2, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2
OFFSET
1,6
COMMENTS
First differs from A293439 at n = 128.
LINKS
Eric Weisstein's World of Mathematics, Unitary Divisor.
FORMULA
Additive with a(p^e) = A209229(e).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} (P(2^k) - P(2^k+1)) = -0.13145993422430119364..., where P(s) is the prime zeta function.
MATHEMATICA
f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(x -> (x == 1 << valuation(x, 2)), factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 28 2023
STATUS
approved