A349693 Dirichlet convolution of the ruler function (A001511) with itself.

1, 4, 2, 10, 2, 8, 2, 20, 3, 8, 2, 20, 2, 8, 4, 35, 2, 12, 2, 20, 4, 8, 2, 40, 3, 8, 4, 20, 2, 16, 2, 56, 4, 8, 4, 30, 2, 8, 4, 40, 2, 16, 2, 20, 6, 8, 2, 70, 3, 12, 4, 20, 2, 16, 4, 40, 4, 8, 2, 40, 2, 8, 6, 84, 4, 16, 2, 20, 4, 16, 2, 60, 2, 8, 6, 20, 4, 16, 2, 70

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

Dirichlet g.f.: zeta(s)^2 * 4^s / (2^s-1)^2.

a(n) = Sum_{d|n} A001511(d) * A001511(n/d).

a(n) = Sum_{d|n} A000217(A001511(d)).

Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - Amiram Eldar, Nov 25 2021

Sum_{k=1..n} a(k) ~ 4*n*(log(n) - 1 + 2*gamma - 2*log(2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Nov 26 2021

Maple

a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[
    divisors](n)))(k-> padic[ordp](2*k, 2)):
seq(a(n), n=1..80);  # Alois P. Heinz, Nov 25 2021

Mathematica

Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]
f[p_, e_] := If[p == 2, Binomial[e + 3, 3], e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 80] (* Amiram Eldar, Nov 25 2021 *)

Prog

(PARI)
A001511(n) = (1+valuation(n, 2));
A349693(n) = sumdiv(n, d, A001511(n/d)*A001511(d)); \\ Antti Karttunen, Nov 25 2021
(Python)
from sympy import divisor_count
def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # Chai Wah Wu, Jul 13 2022

Crossrefs

Cf. A000217, A001511, A115364, A129628.

Sequence in context: A052915 A130273 A016516 * A247414 A138569 A191725

Adjacent sequences: A349690 A349691 A349692 * A349694 A349695 A349696

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Nov 25 2021

Status

approved