This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349693 #25 Jul 28 2025 18:02:20
%S A349693 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 A349693 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 A349693 40,2,8,6,84,4,16,2,20,4,16,2,60,2,8,6,20,4,16,2,70
%N A349693 Dirichlet convolution of the ruler function (A001511) with itself.
%C A349693 Dirichlet convolution of A000005 with A104117. - _Ridouane Oudra_, Jul 23 2025
%H A349693 Antti Karttunen, <a href="/A349693/b349693.txt">Table of n, a(n) for n = 1..20000</a>
%F A349693 Dirichlet g.f.: zeta(s)^2 * 4^s / (2^s-1)^2.
%F A349693 a(n) = Sum_{d|n} A001511(d) * A001511(n/d).
%F A349693 a(n) = Sum_{d|n} A000217(A001511(d)).
%F A349693 Multiplicative with a(p^e) = binomial(e+3,3) if p = 2 and e+1 otherwise. - _Amiram Eldar_, Nov 25 2021
%F A349693 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
%F A349693 From _Ridouane Oudra_, Jul 23 2025: (Start)
%F A349693 a(n) = Sum_{i=0..A007814(n)} (i+1)*tau(n/2^i).
%F A349693 a(n) = Sum_{d|n} A115364(d).
%F A349693 a(n) = (1/6)*A090739(n)*A085058(n-1)*A000005(n).
%F A349693 a(n) = (1/6)*A001511(n)*A090739(n)*A099777(n).
%F A349693 a(n) = (1/3)*A115364(n)*A372784(n).
%F A349693 a(n) = A001227(n)*A000292(A001511(n)).
%F A349693 a(2*n+1) = tau(2*n+1).
%F A349693 a(2^k*(2*n+1)) = binomial(k+3, 3)*tau(2*n+1), for k, n >= 0. (End)
%p A349693 a:= n-> (f-> add(f(d)*f(n/d), d=numtheory[divisors](n)))(k-> padic[ordp](2*k, 2)):
%p A349693 seq(a(n), n=1..80); # _Alois P. Heinz_, Nov 25 2021
%t A349693 Table[Sum[IntegerExponent[2 d, 2] IntegerExponent[2 n/d, 2], {d, Divisors[n]}], {n, 1, 80}]
%t A349693 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 A349693 (PARI)
%o A349693 A001511(n) = (1+valuation(n,2));
%o A349693 A349693(n) = sumdiv(n,d,A001511(n/d)*A001511(d)); \\ _Antti Karttunen_, Nov 25 2021
%o A349693 (Python)
%o A349693 from sympy import divisor_count
%o A349693 def A349693(n): return divisor_count(n)*(m:=(n&-n).bit_length()+1)*(m+1)//6 # _Chai Wah Wu_, Jul 13 2022
%Y A349693 Cf. A000217, A001511, A115364, A129628.
%Y A349693 Cf. A000005, A104117, A007814, A085058, A090739, A099777, A372784, A001227, A000292,
%K A349693 nonn,mult
%O A349693 1,2
%A A349693 _Ilya Gutkovskiy_, Nov 25 2021