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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.

%I A342323 #20 Feb 09 2024 10:36:22
%S A342323 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,
%T A342323 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,
%U A342323 1,1,1,1,1,1,1,1,11,1,1,2,3,2,1,1,1,2,3,1,1,1,1
%N 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.
%C A342323 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.
%H A342323 Jianing Song, <a href="/A342323/b342323.txt">Table of n, a(n) for n = 0..5049</a> (the first 100 antidiagonals)
%F A342323 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.
%e A342323 Table begins:
%e A342323   n\k |  1  2  3  4  5  6  7  8  9 10 11 12
%e A342323   ------------------------------------------
%e A342323     0 |  1  1  1  1  1  1  1  1  1  1  1  1
%e A342323     1 |  1  2  3  2  5  1  7  2  3  1 11  1
%e A342323     2 |  1  1  1  1  1  3  1  1  1  1  1  1
%e A342323     3 |  1  2  1  2  1  1  1  2  1  1  1  1
%e A342323     4 |  1  1  3  1  1  1  1  1  3  5  1  1
%e A342323     5 |  1  2  1  2  1  3  1  2  1  1  1  1
%e A342323     6 |  1  1  1  1  5  1  1  1  1  1  1  1
%e A342323     7 |  1  2  3  2  1  1  1  2  3  1  1  1
%e A342323     8 |  1  1  1  1  1  3  7  1  1  1  1  1
%e A342323     9 |  1  2  1  2  1  1  1  2  1  5  1  1
%e A342323    10 |  1  1  3  1  1  1  1  1  3  1  1  1
%e A342323    11 |  1  2  1  2  5  3  1  2  1  1  1  1
%e A342323    12 |  1  1  1  1  1  1  1  1  1  1 11  1
%t A342323 A342323[n_, k_] := GCD[k, Cyclotomic[k, n]];
%t A342323 Table[A342323[n-k+1, k], {n, 0, 15}, {k, n+1}] (* _Paolo Xausa_, Feb 09 2024 *)
%o A342323 (PARI) T(n,k) = gcd(k, polcyclo(k,n))
%Y A342323 Cf. A342255.
%K A342323 nonn,easy,tabl
%O A342323 0,5
%A A342323 _Jianing Song_, Mar 08 2021