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%I #21 Oct 03 2023 13:11:49
%S 1,1,1,7,1,1,1,5,1,1,1,7,1,1,1,31,1,1,1,7,1,1,1,5,1,1,1,7,1,1,1,21,1,
%T 1,1,7,1,1,1,5,1,1,1,7,1,1,1,31,1,1,1,7,1,1,1,5,1,1,1,7,1,1,1,127,1,1,
%U 1,7,1,1,1,5,1,1,1,7,1,1,1,31,1,1,1,7,1,1,1,5,1,1,1,7,1,1,1,21,1,1,1,7,1,1
%N Denominator of sigma(4n)/sigma(n).
%H Antti Karttunen, <a href="/A088840/b088840.txt">Table of n, a(n) for n = 1..16384</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F From _Amiram Eldar_, Oct 03 2023: (Start)
%F Multiplicative with a(2^e) = (((-1)^e+2)*(2^(e+1)-1))/3 = A213243(e+1), and a(p^e) = 1 for an odd prime p.
%F a(n) = A213243(A007814(n+1)).
%F Dirichlet g.f.: ((8^s + 4^s + 2^(s+1))/(8^s + 4^s - 2^(s+2) - 4)) * zeta(s).
%F Sum_{k=1..n} a(k) = (2*n/(3*log(2))) * (log(n) + gamma - 1 + 7*log(2)/12), where gamma is Euler's constant (A001620). (End)
%t Table[Denominator[DivisorSigma[1, 4*n]/DivisorSigma[1, n]], {n, 1, 128}]
%t a[n_] := Module[{e = IntegerExponent[n, 2]}, (((-1)^e+2)*(2^(e+1)-1))/3]; Array[a, 100] (* _Amiram Eldar_, Oct 03 2023 *)
%o (PARI) A088840(n) = denominator(sigma(4*n)/sigma(n)); \\ _Antti Karttunen_, Nov 18 2017
%o (PARI) a(n) = {my(e = valuation(n, 2)); (((-1)^e+2) * (2^(e+1)-1))/3;} \\ _Amiram Eldar_, Oct 03 2023
%Y See A088839 for numerator.
%Y Cf. A088837, A088838, A088841, A038712, A080278, A000203, A001620, A007814, A193553, A213243.
%K easy,nonn,mult,frac
%O 1,4
%A _Labos Elemer_, Nov 04 2003
%E Typo in definition corrected by _Antti Karttunen_, Nov 18 2017
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