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A370565 Size of the group Q_3*/(Q_3*)^n, where Q_3 is the field of 3-adic numbers. 6
1, 4, 9, 8, 5, 36, 7, 16, 81, 20, 11, 72, 13, 28, 45, 32, 17, 324, 19, 40, 63, 44, 23, 144, 25, 52, 729, 56, 29, 180, 31, 64, 99, 68, 35, 648, 37, 76, 117, 80, 41, 252, 43, 88, 405, 92, 47, 288, 49, 100, 153, 104, 53, 2916, 55, 112, 171, 116, 59, 360, 61, 124, 567, 128 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
We have Q_3* = 3^Z X Z_3*, so Q_3*/(Q_3*)^k = (3^Z/3^(kZ)) X (Z_p*/(Z_3*)^k). Note that 3^Z/3^(kZ) is a cyclic group of order k. For the group structure of (Z_3*/(Z_3*)^k), see A370050.
LINKS
FORMULA
Write n = 3^e * n' with k' not being divisible by 3, then a(n) = n * 3^e * gcd(2,n').
Multiplicative with a(3^e) = 3^(2*e), a(2^e) = 2^(e+1) and a(p^e) = p^e for primes p != 2, 3.
a(n) = n * A370180(n).
From Amiram Eldar, May 20 2024: (Start)
Dirichlet g.f.: ((1 + 1/2^(s-1)) * (1 - 1/3^(s-1))/(1 - 1/3^(s-2))) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (n^2/(2*log(3))) * (log(n) + gamma - 1/2 + log(3) - log(2)/3), where gamma is Euler's constant (A001620). (End)
MATHEMATICA
a[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, 2^Min[e2, 1] * 3^e3 * n]; Array[a, 100] (* Amiram Eldar, May 20 2024 *)
PROG
(PARI) a(n, {p=3}) = my(e = valuation(n, p)); n * p^e*gcd(p-1, n/p^e)
CROSSREFS
Cf. A370180.
Sequence in context: A134902 A110992 A199203 * A370567 A371500 A197580
KEYWORD
nonn,easy,mult
AUTHOR
Jianing Song, Apr 30 2024
STATUS
approved

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Last modified July 24 22:37 EDT 2024. Contains 374585 sequences. (Running on oeis4.)