cp's OEIS Frontend

A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

%I A318511 #15 May 30 2025 01:10:05
%S A318511 1,2,9,2,25,9,49,4,27,25,121,9,169,49,225,6,289,27,361,25,441,121,529,
%T A318511 18,-125,169,405,49,841,225,961,12,1089,289,1225,27,1369,361,1521,50,
%U A318511 1681,441,1849,121,675,529,2209,27,-1029,-125,2601,169,2809,405,3025,98,3249,841,3481,225,3721,961,1323,20,4225,1089,4489,289,4761,1225,5041,27
%N A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
%C A318511 No zeros among the first 2^20 terms.
%C A318511 For odd primes p, it seems that a(p) = p^2.
%H A318511 Antti Karttunen, <a href="/A318511/b318511.txt">Table of n, a(n) for n = 1..16384</a>
%H A318511 Vaclav Kotesovec, <a href="/A318511/a318511.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A318511 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064549(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318511 From _Vaclav Kotesovec_, May 10 2025: (Start)
%F A318511 Let f(s) = Product_{p prime} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
%F A318511 Sum_{k=1..n} A318511(k) / A318512(k) ~ n^3 * sqrt(Pi*f(3)/(54*log(n))) * (1 + (1/3 - gamma/2 - f'(3)/(2*f(3)) - 3*zeta'(2)/Pi^2) / (2*log(n))), where
%F A318511 f(3) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144...
%F A318511 f'(3)/f(3) = Sum_{p prime} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 1.469536740824614833203393993450164364663334798759143895712...
%F A318511 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A318511 f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 10 2025 *)
%o A318511 (PARI)
%o A318511 up_to = 65537;
%o A318511 A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
%o A318511 DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};
%o A318511 v318511_12 = DirSqrt(vector(up_to, n, A064549(n)));
%o A318511 A318511(n) = numerator(v318511_12[n]);
%Y A318511 Cf. A064549, A318512 (denominators).
%Y A318511 Cf. also A317935.
%K A318511 sign,frac
%O A318511 1,2
%A A318511 _Antti Karttunen_, Aug 30 2018