This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370565 #18 May 20 2024 02:30:57
%S A370565 1,4,9,8,5,36,7,16,81,20,11,72,13,28,45,32,17,324,19,40,63,44,23,144,
%T A370565 25,52,729,56,29,180,31,64,99,68,35,648,37,76,117,80,41,252,43,88,405,
%U A370565 92,47,288,49,100,153,104,53,2916,55,112,171,116,59,360,61,124,567,128
%N A370565 Size of the group Q_3*/(Q_3*)^n, where Q_3 is the field of 3-adic numbers.
%C A370565 We have Q_3* = 3^Z X Z_3*, so Q_3*/(Q_3*)^k = (3^Z/3^(kZ)) X (Z_p*/(Z_3*)^k). Note that 3^Z/3^(kZ) is a cyclic group of order k. For the group structure of (Z_3*/(Z_3*)^k), see A370050.
%H A370565 Jianing Song, <a href="/A370565/b370565.txt">Table of n, a(n) for n = 1..10000</a>
%F A370565 Write n = 3^e * n' with k' not being divisible by 3, then a(n) = n * 3^e * gcd(2,n').
%F A370565 Multiplicative with a(3^e) = 3^(2*e), a(2^e) = 2^(e+1) and a(p^e) = p^e for primes p != 2, 3.
%F A370565 a(n) = n * A370180(n).
%F A370565 From _Amiram Eldar_, May 20 2024: (Start)
%F A370565 Dirichlet g.f.: ((1 + 1/2^(s-1)) * (1 - 1/3^(s-1))/(1 - 1/3^(s-2))) * zeta(s-1).
%F A370565 Sum_{k=1..n} a(k) ~ (n^2/(2*log(3))) * (log(n) + gamma - 1/2 + log(3) - log(2)/3), where gamma is Euler's constant (A001620). (End)
%t A370565 a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, 2^Min[e2, 1] * 3^e3 * n]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)
%o A370565 (PARI) a(n, {p=3}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)
%Y A370565 Row 2 of A370067. Cf. A001620, A370050, A370564, A370566, A370567.
%Y A370565 Cf. A370180.
%K A370565 nonn,easy,mult
%O A370565 1,2
%A A370565 _Jianing Song_, Apr 30 2024