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%I #21 Nov 10 2023 10:39:46
%S 0,0,0,-3,0,3,0,0,-6,-5,-3,0,3,5,6,0,0,-9,-7,-6,-3,0,3,6,7,9,0,-10,-5,
%T 0,5,10,-12,-9,-6,-3,0,3,6,9,12,0,0,-15,-12,-11,-9,-6,-3,0,3,6,9,11,
%U 12,15,-15,-14,-10,-7,-5,0,5,7,10,14,15,0,-18,-15,-13,-12,-9,-6,-3,0,3,6,9,12,13,15,18,0
%N Irregular triangle read by rows: row n lists the numbers k such that -n <= k <= n and gcd(k, 2*n+1) != 1.
%C Row n contains 2*n+1 - phi(2*n+1) = A053193(n) terms. Row n has just one term (namely 0) if 2*n+1 is prime.
%H Paolo Xausa, <a href="/A300288/b300288.txt">Table of n, a(n) for n = 1..11968</a> (rows 1..250 of the triangle, flattened)
%F T(n,k) = A299714(n,k) - n.
%e Triangle starts:
%e [01]: 0,
%e [02]: 0,
%e [03]: 0,
%e [04]: -3, 0, 3,
%e [05]: 0,
%e [06]: 0,
%e [07]: -6, -5, -3, 0, 3, 5, 6,
%e [08]: 0,
%e [09]: 0,
%e [10]: -9, -7, -6, -3, 0, 3, 6, 7, 9,
%e [11]: 0,
%e [12]: -10, -5, 0, 5, 10,
%e [13]: -12, -9, -6, -3, 0, 3, 6, 9, 12,
%e [14]: 0,
%e [15]: 0,
%e [16]: -15, -12, -11, -9, -6, -3, 0, 3, 6, 9, 11, 12, 15,
%e [17]: -15, -14, -10, -7, -5, 0, 5, 7, 10, 14, 15,
%e [18]: 0,
%e [19]: -18, -15, -13, -12, -9, -6, -3, 0, 3, 6, 9, 12, 13, 15, 18,
%e [20]: 0,
%e ...
%t A300288row[n_]:=With[{q=2n+1},If[PrimeQ[q],{0},Select[Range[-n, n],GCD[#, q]!=1&]]];Array[A300288row,20] (* _Paolo Xausa_, Nov 10 2023 *)
%o (PARI) is(n, k)= ( gcd(k, 2*n+1)!=1 );
%o for (n=1, 33, for (k=-n, +n, if (is(n, k), print1(k, ", ") ); ); );
%Y Cf. A053193, A299714.
%K sign,tabf
%O 1,4
%A _Joerg Arndt_, Mar 02 2018
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