A318440 a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.

0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 4, 2, 1, 2, 1, 3, 2, 1, 2, 4, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 1, 4, 3, 3, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 1, 1, 4, 4, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 2, 1, 4, 7, 1, 1, 3, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 2, 4, 1, 3, 4, 4, 1, 2, 1, 2, 3

Offset

1,9

Comments

After two initial terms, all terms are positive.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = A046645(n) - A007814(n).

a(n) = A007814(A299150(n)).

Additive with a(p^e) = (1 + (p mod 2))*e - A000120(e). - Amiram Eldar, Apr 28 2023

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -1 + Sum_{p prime} f(1/p) = 0.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 30 2023

Mathematica

f[p_, e_] := (1 + Mod[p, 2])*e - DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

Prog

(PARI)
A007814(n) = valuation(n, 2);
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e), factor(n)[, 2]));
A318440(n) = A046645(n) - A007814(n);

Crossrefs

Cf. A000120, A007814, A046645, A077761, A299150, A318656.

Cf. also A305439.

Sequence in context: A099501 A089762 A257567 * A307790 A189965 A258820

Adjacent sequences: A318437 A318438 A318439 * A318441 A318442 A318443

Keyword

nonn,easy

Author

Antti Karttunen, Sep 02 2018

Status

approved