login
A349693
Dirichlet convolution of the ruler function (A001511) with itself.
1
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
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
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Nov 25 2021
STATUS
approved