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A088839 Numerator of sigma(4n)/sigma(n).

%I A088839 #21 Jan 06 2023 06:33:30
%S A088839 7,5,7,31,7,5,7,21,7,5,7,31,7,5,7,127,7,5,7,31,7,5,7,21,7,5,7,31,7,5,
%T A088839 7,85,7,5,7,31,7,5,7,21,7,5,7,31,7,5,7,127,7,5,7,31,7,5,7,21,7,5,7,31,
%U A088839 7,5,7,511,7,5,7,31,7,5,7,21,7,5,7,31,7,5,7,127,7,5,7,31,7,5,7,21,7,5,7,31
%N A088839 Numerator of sigma(4n)/sigma(n).
%H A088839 Antti Karttunen, <a href="/A088839/b088839.txt">Table of n, a(n) for n = 1..16384</a>
%H A088839 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A088839 a(n) = (8*A006519(n)-1)/(1+2*A096268(n)). - _Robert Israel_, Nov 19 2017
%F A088839 From _Amiram Eldar_, Jan 06 2023: (Start)
%F A088839 a(n) = numerator(A193553(n)/A000203(n)).
%F A088839 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A088840(k) = 3*A065442 + 1 = 5.820085... . (End)
%p A088839 f:= proc(n) local m;
%p A088839   m:= padic:-ordp(n,2);
%p A088839   if m::odd then (2^(m+3)-1)/3 else 2^(m+3)-1 fi
%p A088839 end proc:
%p A088839 map(f, [$1..200]); # _Robert Israel_, Nov 19 2017
%t A088839 k=4; Table[Numerator[DivisorSigma[1, k*n]/DivisorSigma[1, n]], {n, 1, 128}]
%o A088839 (PARI) A088839(n) = numerator(sigma(4*n)/sigma(n)); \\ _Antti Karttunen_, Nov 18 2017
%Y A088839 For denominator see A088840.
%Y A088839 Cf. A000203, A038712, A088837, A088838, A088841, A080278, A193553.
%Y A088839 Cf. A006519, A065442, A096268.
%K A088839 easy,nonn,frac
%O A088839 1,1
%A A088839 _Labos Elemer_, Nov 04 2003
%E A088839 Typo in definition corrected by _Antti Karttunen_, Nov 18 2017