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A202137 Numbers k such that 24k + 1 is neither square nor prime.

%I A202137 #36 Sep 08 2022 08:46:01
%S A202137 6,9,11,16,20,21,23,27,29,30,31,33,34,36,37,38,41,44,45,46,49,53,56,
%T A202137 58,59,60,61,63,64,65,66,68,71,72,76,79,80,81,82,85,86,91,93,94,96,97,
%U A202137 98,101,102,104,106,107,110,111,114,115,116,120,121,122,124
%N A202137 Numbers k such that 24k + 1 is neither square nor prime.
%C A202137 Conjecture: sequence contains arbitrarily long runs of consecutive integers.
%C A202137 First runs with lengths 1..4 are 6; 20, 21; 29, 30, 31; 58, 59, 60, 61.
%C A202137 Records in run lengths are 1, 2, 3, 4, 5, 6, 7, 9, 13, 17, 20, 23, 32, 33, 36, 40, 41, 43, 48, 49, 52, 69, 77, 89, 97, 99, 108, 126, 135, 148, 149
%C A202137 with corresponding first terms of runs: 6, 20, 29, 58, 148, 163, 378, 449, 936, 1675, 5740, 7075, 15915, 35545, 112303, 229944, 469454, 628921, 775480, 902518, 1003826, 1208039, 12542948, 29223210, 33015691, 224430268, 260333109, 530363391, 3713119689, 7962252405, 9312173798.
%C A202137 Conjecture is easy to prove using the Chinese Remainder Theorem and the fact that the gaps between squares grow. - _Robert Israel_, Jan 25 2018
%H A202137 Robert Israel, <a href="/A202137/b202137.txt">Table of n, a(n) for n = 1..10000</a>
%p A202137 filter:= n -> not issqr(24*n+1) and not isprime(24*n+1):
%p A202137 select(filter, [$1..200]); # _Robert Israel_, Jan 25 2018
%t A202137 Select[Range[150],!PrimeQ[24#+1]&&!IntegerQ[Sqrt[24#+1]]&] (* _Harvey P. Dale_, Dec 01 2015 *)
%o A202137 (PARI) for(n=1,200,m=24*n+1;if(isprime(m)+issquare(m),,print1(n",")))
%o A202137 (Magma) [n: n in [1..200] | not IsSquare(24*n+1) and not IsPrime(24*n+1)]; // _Vincenzo Librandi_, Jan 26 2018
%Y A202137 Cf. A089237 (list of primes and squares), A089229 (neither primes nor squares).
%K A202137 nonn
%O A202137 1,1
%A A202137 _Zak Seidov_, Dec 15 2011