A342255 Square array read by ascending antidiagonals: T(n,k) = gcd(k, Phi_k(n)), where Phi_k is the k-th cyclotomic polynomial, n >= 1, k >= 1.

1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 5, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 3, 7, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 5, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 2, 3, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 13

Offset: 1

Author

Jianing Song, Mar 07 2021

Keywords

• nonn
• easy
• tabl

Comments

T(n,k) is either 1 or a prime.

Since p is a prime factor of Phi_k(n) => either p == 1 (mod k) or p is the largest prime factor of k. As a result, T(n,k) = 1 if and only if all prime factors of Phi_k(n) are congruent to 1 modulo k.

Examples

			Table begins
  n\k |  1  2  3  4  5  6  7  8  9 10 11 12
  ------------------------------------------
    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 = 1..5050 (the first 100 antidiagonals)
• Jianing Song, Proof for the first formula

Crossrefs

Cf. A253240, A323748, A014963 (row 1), A253235 (indices of columns with only 1), A342256 (indices of columns with some elements > 1), A342257 (period of each column, also maximum value of each column), A013595 (coefficients of cyclotomic polynomials).

A342323 is the same table with offset 0.

Programs

Mathematica

A342255[n_, k_] := GCD[k, Cyclotomic[k, n]];
Table[A342255[n-k+1,k], {n, 15}, {k, n}] (* 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. See my link above for the proof.

T(n,k) = T(n,k*p^a) for all a, where p is the largest prime factor of k.

T(n,k) = Phi_k(n)/A323748(n,k) for n >= 2, k != 2.

For prime p, T(n,p^e) = p if n == 1 (mod p), 1 otherwise.

For odd prime p, T(n,2*p^e) = p if n == -1 (mod p), 1 otherwise.