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%I #36 Sep 08 2022 08:46:01
%S 6,9,11,16,20,21,23,27,29,30,31,33,34,36,37,38,41,44,45,46,49,53,56,
%T 58,59,60,61,63,64,65,66,68,71,72,76,79,80,81,82,85,86,91,93,94,96,97,
%U 98,101,102,104,106,107,110,111,114,115,116,120,121,122,124
%N Numbers k such that 24k + 1 is neither square nor prime.
%C Conjecture: sequence contains arbitrarily long runs of consecutive integers.
%C First runs with lengths 1..4 are 6; 20, 21; 29, 30, 31; 58, 59, 60, 61.
%C 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 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 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 Robert Israel, <a href="/A202137/b202137.txt">Table of n, a(n) for n = 1..10000</a>
%p filter:= n -> not issqr(24*n+1) and not isprime(24*n+1):
%p select(filter, [$1..200]); # _Robert Israel_, Jan 25 2018
%t Select[Range[150],!PrimeQ[24#+1]&&!IntegerQ[Sqrt[24#+1]]&] (* _Harvey P. Dale_, Dec 01 2015 *)
%o (PARI) for(n=1,200,m=24*n+1;if(isprime(m)+issquare(m),,print1(n",")))
%o (Magma) [n: n in [1..200] | not IsSquare(24*n+1) and not IsPrime(24*n+1)]; // _Vincenzo Librandi_, Jan 26 2018
%Y Cf. A089237 (list of primes and squares), A089229 (neither primes nor squares).
%K nonn
%O 1,1
%A _Zak Seidov_, Dec 15 2011