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

Author

Jianing Song, Apr 30 2024

Keywords

• nonn
• tabl
• easy

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.

Examples

			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
		

Links

• Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)

Crossrefs

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

Programs

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)

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).