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Size of the group Z_7*/(Z_7*)^n, where Z_7 is the ring of 7-adic integers.
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%I #19 May 20 2024 02:27:40

%S 1,2,3,2,1,6,7,2,3,2,1,6,1,14,3,2,1,6,1,2,21,2,1,6,1,2,3,14,1,6,1,2,3,

%T 2,7,6,1,2,3,2,1,42,1,2,3,2,1,6,49,2,3,2,1,6,1,14,3,2,1,6,1,2,21,2,1,

%U 6,1,2,3,14,1,6,1,2,3,2,7,6,1,2,3,2,1,42,1,2,3,2,1,6

%N Size of the group Z_7*/(Z_7*)^n, where Z_7 is the ring of 7-adic integers.

%C We have that Z_7*/(Z_7*)^n is the inverse limit of (Z/7^iZ)*/((Z/7^iZ)*)^n as i tends to infinity. Write n = 7^e * n' with n' not being divisible by 7, then the group is cyclic of order 7^e * gcd(7,n'). See A370050.

%H Jianing Song, <a href="/A370182/b370182.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(7^e) = 7^e, a(2^e) = 2, a(3^e) = 3 and a(p^e) = 1 for primes p != 2, 3, 7.

%F From _Amiram Eldar_, May 20 2024: (Start)

%F Dirichlet g.f.: (1 + 1/2^s) * (1 + 2/3^s) * ((1 - 1/7^s)/(1 - 1/7^(s-1))) * zeta(s).

%F Sum_{k=1..n} a(k) ~ (15*n/(7*log(7))) * (log(n) + gamma - 1 + 2*log(7)/3 - 2*log(3)/5 - log(2)/3), where gamma is Euler's constant (A001620). (End)

%e We have Z_7*/(Z_7*)^7 = Z_7* / ((1+49Z_7) U (18+49Z_7) U (19+49Z_7) U (30+49Z_7) U (31+49Z_7) U (48+49Z_7)) = (Z/49Z)*/((1+49Z) U (18+49Z) U (19+49Z) U (30+49Z) U (31+49Z) U (48+49Z)) = C_7, so a(7) = 7.

%e We have Z_7*/(Z_7*)^14 = Z_7* / ((1+49Z_7) U (18+49Z_7) U (30+49Z_7)) = (Z/49Z)*/((1+49Z) U (18+49Z) U (30+49Z)) = C_14, so a(14) = 14.

%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]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)

%o (PARI) a(n,{p=7}) = my(e = valuation(n, p)); p^e*gcd(p-1, n/p^e)

%Y Row 4 of A370050. Cf. A001620, A297402, A370180, A370181.

%Y Cf. A370567.

%K nonn,easy,mult

%O 1,2

%A _Jianing Song_, Apr 30 2024