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A125212 Numbers m such that no prime exists of the form |k! - m|; or A125211(m) = 0.

%I A125212 #17 Dec 16 2024 19:14:35
%S A125212 1,2,10,16,28,34,36,40,46,50,51,52,56,57,58,64,66,70,76,78,82,86,87,
%T A125212 88,92,93,94,96,100,106,112,116,120,124,126,130,134,135,136,142,144,
%U A125212 146,147,148,154,156,160,162,166,170,171,172,176,177,178,184,186,188,189
%N A125212 Numbers m such that no prime exists of the form |k! - m|; or A125211(m) = 0.
%C A125212 Note the triples of consecutive zeros in A125211 for n = {{50,51,52}, {56,57,58}, {86,87,88}, {92,93,94}, ...}. Most zeros in A125211 have even indices. The middle index of most consecutive zero triples in A125211 is odd and is a multiple of 3. Numbers n such that no prime exists of the form (k! - 3n - 1), (k! - 3n), (k! - 3n + 1) are listed in A125213. The first pair of odd middle indices of zero triples that are not divisible by 3 is n = 325 and n = 329. They belong to the first septuplet of consecutive zeros in A125211. A125211(n) = 0 for 7 consecutive terms from n = 324 to n = 330.
%H A125212 Amiram Eldar, <a href="/A125212/b125212.txt">Table of n, a(n) for n = 1..1000</a>
%e A125212 A125211 begins {0,0,2,3,2,1,3,2,2,0,5,1,7,1,1,0,9,1,6,1,2,1,4,1,2,1,1,0,5,1,8,1,1,0,2,0,10,1,1,0,6,1,10,1,1,0,10,1,3,0,0,0,7,...}.
%e A125212 Thus a(1) = 1, a(2) = 2, a(3) = 10, a(10)-a(12) = {50,51,52}.
%t A125212 k={};Do[If[Length[Select[Range[m],PrimeQ[#!-m]&]]==0,AppendTo[k,m]],{m,189}];k (* _James C. McMahon_, Dec 16 2024 *)
%o A125212 (PARI) isok(m)={for(k=1, m-1, if(isprime(abs(k!-m)), return(0))); 1} \\ _Andrew Howroyd_, Dec 16 2024
%Y A125212 Cf. A125162 (number of primes of the form k! + n).
%Y A125212 Cf. A125163 (numbers n such that no prime exists of the form k! + n).
%Y A125212 Cf. A125164 (numbers n such that no prime exists of the form (k! + 3n - 1), (k! + 3n), (k! + 3n + 1)).
%Y A125212 Cf. A125211, A125213.
%K A125212 nonn
%O A125212 1,2
%A A125212 _Alexander Adamchuk_, Nov 23 2006