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A370564 Size of the group Q_2*/(Q_2*)^n, where Q_2 is the field of 2-adic numbers.

%I A370564 #19 May 20 2024 02:31:09
%S A370564 1,8,3,32,5,24,7,128,9,40,11,96,13,56,15,512,17,72,19,160,21,88,23,
%T A370564 384,25,104,27,224,29,120,31,2048,33,136,35,288,37,152,39,640,41,168,
%U A370564 43,352,45,184,47,1536,49,200,51,416,53,216,55,896,57,232,59,480,61,248,63,8192
%N A370564 Size of the group Q_2*/(Q_2*)^n, where Q_2 is the field of 2-adic numbers.
%C A370564 We have Q_2* = 2^Z X Z_2*, so Q_2*/(Q_2*)^k = (2^Z/2^(kZ)) X (Z_p*/(Z_2*)^k). Note that 2^Z/2^(kZ) is a cyclic group of order k. For the group structure of (Z_2*/(Z_2*)^k), see A370050.
%H A370564 Jianing Song, <a href="/A370564/b370564.txt">Table of n, a(n) for n = 1..10000</a>
%F A370564 If n is odd, then a(n) = n. If n = 2^e * n' is even, where n' is odd, then a(n) = n * 2^(e+1).
%F A370564 Multiplicative with a(2^e) = 2^(2*e+1).
%F A370564 a(n) = n * A297402(n).
%F A370564 From _Amiram Eldar_, May 20 2024: (Start)
%F A370564 Dirichlet g.f.: ((1 - 1/2^(s-1)) * (1 + 1/2^(s-2)) / (1 - 1/2^(s-2))) * zeta(s-1).
%F A370564 Sum_{k=1..n} a(k) ~ (n^2/(2*log(2))) * (log(n) + gamma - 1/2 + log(2)), where gamma is Euler's constant (A001620). (End)
%t A370564 a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^If[e == 0, 0, e + 1] * n]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)
%o A370564 (PARI) a(n) = my(e = valuation(n, 2)); n * 2^e * if(e>=1, 2, 1)
%Y A370564 Row 1 of A370067. Cf. A001620, A370050, A370565, A370566, A370567.
%Y A370564 Cf. A297402.
%K A370564 nonn,easy,mult
%O A370564 1,2
%A A370564 _Jianing Song_, Apr 30 2024