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A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

%I A318314 #24 Oct 05 2021 21:07:54
%S A318314 1,2,1,8,1,2,1,16,2,2,1,8,1,2,1,128,1,4,1,8,1,2,1,16,2,2,2,8,1,2,1,
%T A318314 256,1,2,1,16,1,2,1,16,1,2,1,8,2,2,1,128,2,4,1,8,1,4,1,16,1,2,1,8,1,2,
%U A318314 2,1024,1,2,1,8,1,2,1,32,1,2,2,8,1,2,1,128,8,2,1,8,1,2,1,16,1,4,1,8,1,2,1,256,1,4,2,16,1,2,1,16,1
%N A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.
%C A318314 The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
%C A318314 Note that A318314 differs from A318454 at exactly those n where A001227 differs from A068068, the numbers in A038838. - _Antti Karttunen_, Sep 07 2018
%H A318314 Antti Karttunen, <a href="/A318314/b318314.txt">Table of n, a(n) for n = 1..16384</a>
%F A318314 a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318314 a(n) = 2^A318315(n).
%F A318314 From _Antti Karttunen_, Sep 03-07 2018: (Start, conjectured formulas)
%F A318314 a(n) = A006519(n) * A317934(n), thus multiplicative with a(2^e) = 2^A005187(e), a(p^e) = 2^A011371(e) for odd primes p.
%F A318314 Equally, multiplicative with a(p^e) = 2^(((2-A000035(p))*e)-A000120(e)) for all primes p.
%F A318314 (End)
%t A318314 a35[n_] := (1 - (-1)^n)/2;
%t A318314 a120[n_] := DigitCount[n, 2, 1];
%t A318314 a[n_] := Product[{p, e} = pe; 2^(((2 - a35[p])*e) - a120[e]), {pe, FactorInteger[n]}];
%t A318314 a /@ Range[100] (* _Jean-François Alcover_, Sep 19 2019 *)
%o A318314 (PARI)
%o A318314 up_to = 16384;
%o A318314 A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
%o A318314 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}; \\ From A317937.
%o A318314 v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
%o A318314 A318313(n) = numerator(v318313_15[n]);
%o A318314 A318314(n) = denominator(v318313_15[n]);
%Y A318314 Cf. A005187, A011371, A068068, A318313 (numerators), A318315.
%Y A318314 Cf. also A006519, A046644, A299150, A317932, A317934, A317940, A318454, A318662.
%K A318314 nonn,frac,mult
%O A318314 1,2
%A A318314 _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018