A342323 Square array read by ascending antidiagonals: T(n,k) = gcd(k, Phi_k(n)), where Phi_k is the k-th cyclotomic polynomial, n >= 0, k >= 1.

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 5, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 2, 3, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1

Offset: 0

Author

Jianing Song, Mar 08 2021

Keywords

• nonn
• easy
• tabl

Comments

This is the same table as A342255 but with offset 0. Therefore, the resulting sequences as flattened tables are different. The main entry is A342255.

Examples

			Table begins:
  n\k |  1  2  3  4  5  6  7  8  9 10 11 12
  ------------------------------------------
    0 |  1  1  1  1  1  1  1  1  1  1  1  1
    1 |  1  2  3  2  5  1  7  2  3  1 11  1
    2 |  1  1  1  1  1  3  1  1  1  1  1  1
    3 |  1  2  1  2  1  1  1  2  1  1  1  1
    4 |  1  1  3  1  1  1  1  1  3  5  1  1
    5 |  1  2  1  2  1  3  1  2  1  1  1  1
    6 |  1  1  1  1  5  1  1  1  1  1  1  1
    7 |  1  2  3  2  1  1  1  2  3  1  1  1
    8 |  1  1  1  1  1  3  7  1  1  1  1  1
    9 |  1  2  1  2  1  1  1  2  1  5  1  1
   10 |  1  1  3  1  1  1  1  1  3  1  1  1
   11 |  1  2  1  2  5  3  1  2  1  1  1  1
   12 |  1  1  1  1  1  1  1  1  1  1 11  1
		

Links

• Jianing Song, Table of n, a(n) for n = 0..5049 (the first 100 antidiagonals)

Crossrefs

Cf. A342255.

Programs

Mathematica

A342323[n_, k_] := GCD[k, Cyclotomic[k, n]];
Table[A342323[n-k+1, k], {n, 0, 15}, {k, n+1}] (* Paolo Xausa, Feb 09 2024 *)

PARI

T(n,k) = gcd(k, polcyclo(k,n))

Formula

For k > 1, let p be the largest prime factor of k, then T(n,k) = p if p does not divide n and k = p^e*ord(p,n) for some e > 0, where ord(p,n) is the multiplicative order of n modulo p.