A366074 - OEIS

%I #9 Sep 29 2023 04:06:13

%S 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,

%T 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,

%U 2,3,1,1,1,2,2,2,2,3,1,2,1,2,1,3,2,2,2

%N The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

%C First differs from A293439 at n = 128.

%H Amiram Eldar, <a href="/A366074/b366074.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.

%F Additive with a(p^e) = A209229(e).

%F 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.

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

%o (PARI) a(n) = vecsum(apply(x -> (x == 1 << valuation(x, 2)), factor(n)[, 2]));

%Y Cf. A050376, A064547, A077610, A077761, A209229, A366073.

%K nonn,easy

%O 1,6

%A _Amiram Eldar_, Sep 28 2023