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%I A318512 #69 May 10 2025 11:54:23
%S A318512 1,1,2,1,2,1,2,1,8,1,2,1,2,1,4,1,2,4,2,1,4,1,2,1,8,1,16,1,2,2,2,1,4,1,
%T A318512 4,4,2,1,4,1,2,2,2,1,16,1,2,1,8,4,4,1,2,8,4,1,4,1,2,2,2,1,16,1,4,2,2,
%U A318512 1,4,2,2,2,2,1,16,1,4,2,2,1,128,1,2,2,4,1,4,1,2,8,4,1,4,1,4,1,2,4,16,4,2,2,2,1,8
%N A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.
%C A318512 These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
%C A318512 From _Antti Karttunen_, Sep 02 2018: (Start)
%C A318512 Proof for the above claim:
%C A318512 This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d<n} r(d) * r(n/d)) for n > 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).
%C A318512 By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:
%C A318512 A318649(n) A318681(n) n*A299149(n)
%C A318512 ---------- = ---------- = ------------
%C A318512 A318512(n) A299150(n) A299150(n)
%C A318512 and
%C A318512 A318511(n) A318680(n) n*A318653(n)
%C A318512 ---------- = ---------- = ------------
%C A318512 Ay(n) A299150(n) A299150(n)
%C A318512 where the leftmost ratios are reduced into their lowest terms.
%C A318512 Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).
%C A318512 (End)
%H A318512 Antti Karttunen, <a href="/A318512/b318512.txt">Table of n, a(n) for n = 1..16384</a>
%H A318512 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>
%F A318512 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1. [Equally, one could use A064549 in place of A000290.]
%F A318512 a(n) = 2^A318513(n).
%F A318512 a(n) = A046644(n)/A318651(n).
%F A318512 a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.
%t A318512 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[Denominator[f[n]], {n, 1, 100}] (* _Vaclav Kotesovec_, May 10 2025 *)
%o A318512 (PARI)
%o A318512 up_to = 65537;
%o A318512 A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); };
%o A318512 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 A318512 v318511_12 = DirSqrt(vector(up_to, n, A064549(n)));
%o A318512 A318512(n) = denominator(v318511_12[n]);
%o A318512 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A318512 Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.
%Y A318512 Cf. A318649, A318511 (numerators).
%K A318512 nonn,frac
%O A318512 1,3
%A A318512 _Antti Karttunen_, Aug 30 2018
%E A318512 The main definition changed, more formulas added by _Antti Karttunen_, Aug 31 2018