A370078 a(n) = log_2(A370077(n)).

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset

1,16

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A007814(A005361(A367168(n))).

Additive with a(p^e) = log_2(e) if e is a power of 2, and 0 otherwise.

a(n) = 0 if and only if n is an exponentially odd number (A268335).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * (P(2^k)-P(2^k+1)) = 0.36616241880640645934..., where P(s) is the prime zeta function .

Mathematica

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

Prog

(PARI) s(n) = my(e = valuation(n, 2)); if(n >> e == 1, e, 0);
a(n) = vecsum(apply(x -> s(x), factor(n)[, 2]));

Crossrefs

Cf. A005361, A007814, A268335, A367168, A370077.

Sequence in context: A323591 A105348 A016406 * A372332 A129182 A116857

Adjacent sequences: A370075 A370076 A370077 * A370079 A370080 A370081

Keyword

nonn,easy

Author

Amiram Eldar, Feb 09 2024

Status

approved