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A370181 Size of the group Z_5*/(Z_5*)^n, where Z_5 is the ring of 5-adic integers.

%I A370181 #19 May 20 2024 02:31:20
%S A370181 1,2,1,4,5,2,1,4,1,10,1,4,1,2,5,4,1,2,1,20,1,2,1,4,25,2,1,4,1,10,1,4,
%T A370181 1,2,5,4,1,2,1,20,1,2,1,4,5,2,1,4,1,50,1,4,1,2,5,4,1,2,1,20,1,2,1,4,5,
%U A370181 2,1,4,1,10,1,4,1,2,25,4,1,2,1,20,1,2,1,4,5,2,1,4,1,10
%N A370181 Size of the group Z_5*/(Z_5*)^n, where Z_5 is the ring of 5-adic integers.
%C A370181 We have that Z_5*/(Z_5*)^n is the inverse limit of (Z/5^iZ)*/((Z/5^iZ)*)^n as i tends to infinity. Write n = 5^e * n' with n' not being divisible by 5, then the group is cyclic of order 5^e * gcd(4,n'). See A370050.
%H A370181 Jianing Song, <a href="/A370181/b370181.txt">Table of n, a(n) for n = 1..10000</a>
%F A370181 Multiplicative with a(5^e) = 5^e, a(2) = 2, a(2^e) = 4 for e >= 2 and a(p^e) = 1 for primes p != 2, 5.
%F A370181 From _Amiram Eldar_, May 20 2024: (Start)
%F A370181 Dirichlet g.f.: (1 + 1/2^s + 1/2^(2*s-1)) * ((1 - 1/5^s)/(1 - 1/5^(s-1))) * zeta(s).
%F A370181 Sum_{k=1..n} a(k) ~ (8*n/(5*log(5))) * (log(n) + gamma - 1 + (3/4)*log(5/2)), where gamma is Euler's constant (A001620). (End)
%e A370181 We have Z_5*/(Z_5*)^5 = Z_5* / ((1+25Z_5) U (7+25Z_5) U (18+25Z_5) U (24+25Z_5)) = (Z/25Z)*/((1+25Z) U (7+25Z) U (18+25Z) U (24+25Z)) = C_5, so a(5) = 5.
%e A370181 We have Z_5*/(Z_5*)^10 = Z_5* / ((1+25Z_5) U (24+25Z_5)) = (Z/25Z)*/((1+25Z) U (25+25Z)) = C_10, so a(10) = 10.
%t A370181 a[n_] := Module[{e2 = IntegerExponent[n, 2], e5 = IntegerExponent[n, 5]}, 2^Min[e2, 2] * 5^e5]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)
%o A370181 (PARI) a(n,{p=5}) = my(e = valuation(n, p)); p^e*gcd(p-1, n/p^e)
%Y A370181 Row 3 of A370050. Cf. A001620, A297402, A370180, A370182.
%Y A370181 Cf. A370566.
%K A370181 nonn,easy,mult
%O A370181 1,2
%A A370181 _Jianing Song_, Apr 30 2024