|
%I #19 May 20 2024 02:30:33
%S 1,4,9,8,5,36,49,16,27,20,11,72,13,196,45,32,17,108,19,40,441,44,23,
%T 144,25,52,81,392,29,180,31,64,99,68,245,216,37,76,117,80,41,1764,43,
%U 88,135,92,47,288,2401,100,153,104,53,324,55,784,171,116,59,360,61,124,1323,128
%N Size of the group Q_7*/(Q_7*)^n, where Q_7 is the field of 7-adic numbers.
%C We have Q_7* = 7^Z X Z_7*, so Q_7*/(Q_7*)^k = (7^Z/7^(kZ)) X (Z_p*/(Z_7*)^k). Note that 7^Z/7^(kZ) is a cyclic group of order k. For the group structure of (Z_7*/(Z_7*)^k), see A370050.
%H Jianing Song, <a href="/A370567/b370567.txt">Table of n, a(n) for n = 1..10000</a>
%F Write n = 7^e * n' with k' not being divisible by 7, then a(n) = n * 7^e * gcd(6,n').
%F Multiplicative with a(7^e) = 7^(2*e), a(2^e) = 2^(e+1), a(3^e) = 3^(e+1) and a(p^e) = p^e for primes p != 2, 3, 7.
%F a(n) = n * A370182(n).
%F From _Amiram Eldar_, May 20 2024: (Start)
%F Dirichlet g.f.: ((1 + 1/2^(s-1)) * (1 + 2/3^(s-1)) * (1 - 1/7^(s-1))/(1 - 1/7^(s-2))) * zeta(s-1).
%F Sum_{k=1..n} a(k) ~ (15*n^2/(14*log(7))) * (log(n) + gamma - 1/2 + 2*log(7)/3 - 2*log(3)/5 - log(2)/3), where gamma is Euler's constant (A001620). (End)
%t a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3], e7 = IntegerExponent[n, 7]}, 2^Min[e2, 1] * 3^Min[e3, 1] * 7^e7 * n]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)
%o (PARI) a(n, {p=7}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)
%Y Row 4 of A370067. Cf. A001620, A370050, A370564, A370565, A370566.
%Y Cf. A370182.
%K nonn,easy,mult
%O 1,2
%A _Jianing Song_, Apr 30 2024
|
