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A370567
Size of the group Q_7*/(Q_7*)^n, where Q_7 is the field of 7-adic numbers.
6
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
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
Cf. A370182.
Sequence in context: A110992 A199203 A370565 * A371500 A197580 A081382
KEYWORD
nonn,easy,mult
AUTHOR
Jianing Song, Apr 30 2024
STATUS
approved