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.

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

Cf. A370050. Row 1-4: A370564, A370565, A370566, A370567.

Sequence in context: A079359 A010156 A197590 * A154190 A019981 A194281

Adjacent sequences: A370064 A370065 A370066 * A370068 A370069 A370070

Keyword

nonn,tabl,easy

Author

Jianing Song, Apr 30 2024

Status

approved