A133370 - OEIS

%I #39 Sep 18 2018 21:51:21

%S 2,3,7,13,23,29,43,47,53,67,71,79,83,89,101,103,107,109,113,127,131,

%T 137,149,157,167,173,199,223,229,239,263,269,277,281,311,313,317,337,

%U 349,353,359,373,383,389,397,401,409,421,449,457,461,467,479,487,491

%N Primes p such that p does not divide any term of the Apery sequence A005259 .

%C Malik and Straub give arguments suggesting that this sequence is infinite. - _N. J. A. Sloane_, Aug 06 2017

%H Robert Price, <a href="/A133370/b133370.txt">Table of n, a(n) for n = 1..758</a>

%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.

%H Amita Malik, <a href="/A133370/a133370.nb">Mathematica notebook for generating this sequence and A260793, A291275-A291284</a>

%H Amita Malik, <a href="/A133370/a133370.pdf">List of all primes up to 10000 in this sequence and in A260793, A291275-A291284, together with Mathematica code.</a>

%H E. Rowland, R. Yassawi, <a href="https://arxiv.org/abs/1310.8635">Automatic congruences for diagonals of rational functions</a>, arXiv preprint arXiv:1310.8635 [math.NT], 2013.

%t NeverDividesLucasSeqQ[a_, p_] := And @@ Table[Mod[a[n], p]>0, {n, 0, p-1}];

%t A3[a_, b_, c_, n_ /; n < 0] = 0;

%t A3[a_, b_, c_, 0] = 1;

%t A3[a_, b_, c_, n_] := A3[a, b, c, n] = (((2n - 1)(a (n-1)^2 + a (n-1) + b)) A3[a, b, c, n-1] - c (n-1)^3 A3[a, b, c, n-2])/n^3;

%t A3[a_, b_, c_, d_, n_ /; n < 0] = 0;

%t Agamma[n_] := A3[17, 5, 1, n];

%t Select[Range[1000], PrimeQ[#] && NeverDividesLucasSeqQ[Agamma, #]&] (* _Jean-François Alcover_, Aug 05 2018, copied from Amita Malik's notebook *)

%Y For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

%K nonn

%O 1,1

%A _Philippe Deléham_, Oct 27 2007

%E Terms a(16) onwards computed by Amita Malik - _N. J. A. Sloane_, Aug 21 2017