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A370067
Square array read by ascending antidiagonals: T(n,k) is the size of the group Q_p*/(Q_p*)^k, where p = prime(n), and Q_p is the field of p-adic numbers.
6
1, 1, 8, 1, 4, 3, 1, 4, 9, 32, 1, 4, 3, 8, 5, 1, 4, 9, 16, 5, 24, 1, 4, 3, 8, 25, 36, 7, 1, 4, 9, 8, 5, 12, 7, 128, 1, 4, 3, 16, 25, 36, 7, 16, 9, 1, 4, 9, 16, 5, 12, 49, 32, 81, 40, 1, 4, 3, 8, 5, 36, 7, 16, 9, 20, 11, 1, 4, 3, 8, 5, 12, 7, 16, 27, 100, 11, 96, 1, 4, 9, 16, 5, 36, 7, 32, 9, 20, 11, 72, 13
OFFSET
1,3
COMMENTS
We have Q_p* = p^Z X Z_p*, so Q_p*/(Q_p*)^k = (p^Z/p^(kZ)) X (Z_p*/(Z_p*)^k). Note that p^Z/p^(kZ) is a cyclic group of order k. For the group structure of (Z_p*/(Z_p*)^k), see A370050.
Each row is multiplicative.
LINKS
Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)
FORMULA
T(n,k) = k * A370050(n,k).
Write k = p^e * k' with k' not being divisible by p, and p = prime(n). If p is odd, then T(n,k) = k * p^e * gcd(p-1,k'). If p = 2 and k is odd, then T(n,k) = k. If p = 2 and k is even, then T(n,k) = k * 2^(e+1).
EXAMPLE
Table reads
1, 8, 3, 32, 5, 24, 7, 128, 9, 40
1, 4, 9, 8, 5, 36, 7, 16, 81, 20
1, 4, 3, 16, 25, 12, 7, 32, 9, 100
1, 4, 9, 8, 5, 36, 49, 16, 27, 20
1, 4, 3, 8, 25, 12, 7, 16, 9, 100
1, 4, 9, 16, 5, 36, 7, 32, 27, 20
1, 4, 3, 16, 5, 12, 7, 64, 9, 20
1, 4, 9, 8, 5, 36, 7, 16, 81, 20
1, 4, 3, 8, 5, 12, 7, 16, 9, 20
1, 4, 3, 16, 5, 12, 49, 32, 9, 20
PROG
(PARI) T(n, k) = my(p = prime(n), e = valuation(k, p)); k * p^e*gcd(p-1, k/p^e) * if(p==2 && e>=1, 2, 1)
CROSSREFS
Sequence in context: A079359 A010156 A197590 * A154190 A019981 A194281
KEYWORD
nonn,tabl,easy
AUTHOR
Jianing Song, Apr 30 2024
STATUS
approved