A370566 Size of the group Q_5*/(Q_5*)^n, where Q_5 is the field of 5-adic numbers.

1, 4, 3, 16, 25, 12, 7, 32, 9, 100, 11, 48, 13, 28, 75, 64, 17, 36, 19, 400, 21, 44, 23, 96, 625, 52, 27, 112, 29, 300, 31, 128, 33, 68, 175, 144, 37, 76, 39, 800, 41, 84, 43, 176, 225, 92, 47, 192, 49, 2500, 51, 208, 53, 108, 275, 224, 57, 116, 59, 1200, 61, 124, 63, 256

Offset

1,2

Comments

We have Q_5* = 5^Z X Z_5*, so Q_5*/(Q_5*)^k = (5^Z/5^(kZ)) X (Z_p*/(Z_5*)^k). Note that 5^Z/5^(kZ) is a cyclic group of order k. For the group structure of (Z_5*/(Z_5*)^k), see A370050.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Formula

Write n = 5^e * n' with k' not being divisible by 5, then a(n) = n * 5^e * gcd(4,n').

Multiplicative with a(5^e) = 5^(2*e), a(2) = 4, a(2^e) = 2^(e+2) for e >= 2 and a(p^e) = p^e for primes p != 2, 5.

a(n) = n * A370181(n).

From Amiram Eldar, May 20 2024: (Start)

Dirichlet g.f.: ((1 + 1/2^(s-1) + 1/2^(2*s-3)) * (1 - 1/5^(s-1))/(1 - 1/5^(s-2))) * zeta(s-1).

Sum_{k=1..n} a(k) ~ (4*n^2/(5*log(5))) * (log(n) + gamma - 1/2 + 3*log(5/2)/4), where gamma is Euler's constant (A001620). (End)

Mathematica

a[n_] := Module[{e2 = IntegerExponent[n, 2], e5 = IntegerExponent[n, 5]}, 2^Min[e2, 2] * 5^e5 * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

Prog

(PARI) a(n, {p=5}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)

Crossrefs

Row 3 of A370067. Cf. A001620, A370050, A370564, A370565, A370567.

Cf. A370181.

Sequence in context: A288067 A038233 A176737 * A046162 A060509 A113203

Adjacent sequences: A370563 A370564 A370565 * A370567 A370568 A370569

Keyword

nonn,easy,mult

Author

Jianing Song, Apr 30 2024

Status

approved