A370564 Size of the group Q_2*/(Q_2*)^n, where Q_2 is the field of 2-adic numbers.

1, 8, 3, 32, 5, 24, 7, 128, 9, 40, 11, 96, 13, 56, 15, 512, 17, 72, 19, 160, 21, 88, 23, 384, 25, 104, 27, 224, 29, 120, 31, 2048, 33, 136, 35, 288, 37, 152, 39, 640, 41, 168, 43, 352, 45, 184, 47, 1536, 49, 200, 51, 416, 53, 216, 55, 896, 57, 232, 59, 480, 61, 248, 63, 8192

Offset: 1

Author

Jianing Song, Apr 30 2024

Keywords

• nonn
• easy
• mult

Comments

We have Q_2* = 2^Z X Z_2*, so Q_2*/(Q_2*)^k = (2^Z/2^(kZ)) X (Z_p*/(Z_2*)^k). Note that 2^Z/2^(kZ) is a cyclic group of order k. For the group structure of (Z_2*/(Z_2*)^k), see A370050.

Links

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

Crossrefs

Row 1 of A370067. Cf. A001620, A370050, A370565, A370566, A370567.

Cf. A297402.

Programs

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^If[e == 0, 0, e + 1] * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)

PARI

a(n) = my(e = valuation(n, 2)); n * 2^e * if(e>=1, 2, 1)

Formula

If n is odd, then a(n) = n. If n = 2^e * n' is even, where n' is odd, then a(n) = n * 2^(e+1).

Multiplicative with a(2^e) = 2^(2*e+1).

a(n) = n * A297402(n).

From Amiram Eldar, May 20 2024: (Start)

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

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