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A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.

%I A317934 #33 May 09 2025 10:24:56
%S A317934 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,8,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,8,1,1,
%T A317934 1,4,1,1,1,2,1,1,1,2,2,1,1,8,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,16,1,1,1,2,
%U A317934 1,1,1,4,1,1,2,2,1,1,1,8,8,1,1,2,1,1,1,2,1,2,1,2,1,1,1,8,1,2,2,4,1,1,1,2,1
%N A317934 Multiplicative with a(p^n) = 2^A011371(n); denominators for certain "Dirichlet Square Roots" sequences.
%C A317934 a(n) is the denominator of certain rational valued sequences f(n), that have been defined as f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)), with f(1) = 1, where b(n) is a sequence like A034444 and A037445.
%C A317934 Many of the same observations as given in A046644 apply also here. Note that A011371 shares with A005187 the property that A011371(x+y) <= A011371(x) + A011371(y), with equivalence attained only when A004198(x,y) = 0, and also the property that A011371(2^(k+1)) = 1 + 2*A011371(2^k).
%C A317934 The following list gives such pairs num(n), b(n) for which b(n) is Dirichlet convolution of num(n)/a(n).
%C A317934   Numerators   Dirichlet convolution of numerator(n)/a(n) yields
%C A317934   -------      -----------
%C A317934   A317933      A034444
%C A317934   A317941      A037445
%C A317934   A317940      A046644
%C A317934 Expansion of Dirichlet g.f. Product_{prime} 1/(1 - 2/p^s)^(1/2) is A046643/A317934. - _Vaclav Kotesovec_, May 08 2025
%H A317934 Antti Karttunen, <a href="/A317934/b317934.txt">Table of n, a(n) for n = 1..65537</a>
%H A317934 Vaclav Kotesovec, <a href="/A317934/a317934.jpg">Graph - the asymptotic ratio (1000000 terms)</a>
%H A317934 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>
%F A317934 a(n) = 2^A317946(n).
%F A317934 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (b(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1, where b is A034444, A037445 or A046644 for example.
%F A317934 Sum_{k=1..n} A046643(k)/a(k) ~ n * sqrt(A167864*log(n)/(Pi*log(2))) * (1 + (4*(gamma - 1) + 5*log(2) - 4*A347195)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 08 2025
%o A317934 (PARI)
%o A317934 A011371(n) = (n - hammingweight(n));
%o A317934 A317934(n) = factorback(apply(e -> 2^A011371(e),factor(n)[,2]));
%o A317934 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-2*X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 07 2025
%o A317934 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A317934 Cf. A011371, A034444, A317940, A317941, A317946.
%Y A317934 Cf. A317933, A317940, A317941 (numerator-sequences).
%Y A317934 Cf. also A046644, A299150, A299152, A317832, A317932, A317926 (for denominator sequences of other similar constructions).
%Y A317934 Cf. A046643, A167864, A347195.
%K A317934 nonn,frac,mult
%O A317934 1,4
%A A317934 _Antti Karttunen_, Aug 12 2018