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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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.
LINKS
Jianing Song, Table of n, a(n) for n = 1..5050 (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. 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.
EXAMPLE
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
PROG
(PARI) T(n, k) = gcd(k, polcyclo(k, n))
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.
Sequence in context: A260533 A107359 A307641 * A112377 A354521 A277760
KEYWORD
nonn,easy,tabl
AUTHOR
Jianing Song, Mar 07 2021
STATUS
approved

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Last modified November 30 23:17 EST 2023. Contains 367463 sequences. (Running on oeis4.)