A370567 Size of the group Q_7*/(Q_7*)^n, where Q_7 is the field of 7-adic numbers.

1, 4, 9, 8, 5, 36, 49, 16, 27, 20, 11, 72, 13, 196, 45, 32, 17, 108, 19, 40, 441, 44, 23, 144, 25, 52, 81, 392, 29, 180, 31, 64, 99, 68, 245, 216, 37, 76, 117, 80, 41, 1764, 43, 88, 135, 92, 47, 288, 2401, 100, 153, 104, 53, 324, 55, 784, 171, 116, 59, 360, 61, 124, 1323, 128

Offset

1,2

Comments

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.

Links

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

Formula

Write n = 7^e * n' with k' not being divisible by 7, then a(n) = n * 7^e * gcd(6,n').

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.

a(n) = n * A370182(n).

From Amiram Eldar, May 20 2024: (Start)

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).

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)

Mathematica

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 *)

Prog

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

Crossrefs

Row 4 of A370067. Cf. A001620, A370050, A370564, A370565, A370566.

Cf. A370182.

Sequence in context: A110992 A199203 A370565 * A371500 A197580 A081382

Adjacent sequences: A370564 A370565 A370566 * A370568 A370569 A370570

Keyword

nonn,easy,mult

Author

Jianing Song, Apr 30 2024

Status

approved