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Dirichlet convolution of the ruler function (A001511) with itself.
1

%I #22 Jul 14 2022 02:19:42

%S 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,

%T 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,

%U 40,2,8,6,84,4,16,2,20,4,16,2,60,2,8,6,20,4,16,2,70

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

%H Antti Karttunen, <a href="/A349693/b349693.txt">Table of n, a(n) for n = 1..20000</a>

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

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

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

%F Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - _Amiram Eldar_, Nov 25 2021

%F 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

%p a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[

%p divisors](n)))(k-> padic[ordp](2*k, 2)):

%p seq(a(n), n=1..80); # _Alois P. Heinz_, Nov 25 2021

%t Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]

%t 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 *)

%o (PARI)

%o A001511(n) = (1+valuation(n,2));

%o A349693(n) = sumdiv(n,d,A001511(n/d)*A001511(d)); \\ _Antti Karttunen_, Nov 25 2021

%o (Python)

%o from sympy import divisor_count

%o def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # _Chai Wah Wu_, Jul 13 2022

%Y Cf. A000217, A001511, A115364, A129628.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Nov 25 2021