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

0,5

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.

Links

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

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A342255.

Sequence in context: A357138 A357180 A196660 * A135222 A285706 A357181

Adjacent sequences: A342320 A342321 A342322 * A342324 A342325 A342326

Keyword

nonn,easy,tabl

Author

Jianing Song, Mar 08 2021

Status

approved