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 A202137 Numbers k such that 24k + 1 is neither square nor prime. 1
 6, 9, 11, 16, 20, 21, 23, 27, 29, 30, 31, 33, 34, 36, 37, 38, 41, 44, 45, 46, 49, 53, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 71, 72, 76, 79, 80, 81, 82, 85, 86, 91, 93, 94, 96, 97, 98, 101, 102, 104, 106, 107, 110, 111, 114, 115, 116, 120, 121, 122, 124 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: sequence contains arbitrarily long runs of consecutive integers. First runs with lengths 1..4 are 6; 20, 21; 29, 30, 31; 58, 59, 60, 61. 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 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. Conjecture is easy to prove using the Chinese Remainder Theorem and the fact that the gaps between squares grow. - Robert Israel, Jan 25 2018 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 MAPLE filter:= n -> not issqr(24*n+1) and not isprime(24*n+1): select(filter, [\$1..200]); # Robert Israel, Jan 25 2018 MATHEMATICA Select[Range[150], !PrimeQ[24#+1]&&!IntegerQ[Sqrt[24#+1]]&] (* Harvey P. Dale, Dec 01 2015 *) PROG (PARI) for(n=1, 200, m=24*n+1; if(isprime(m)+issquare(m), , print1(n", "))) (Magma) [n: n in [1..200] | not IsSquare(24*n+1) and not IsPrime(24*n+1)]; // Vincenzo Librandi, Jan 26 2018 CROSSREFS Cf. A089237 (list of primes and squares), A089229 (neither primes nor squares). Sequence in context: A184104 A315950 A026614 * A302580 A031256 A032730 Adjacent sequences: A202134 A202135 A202136 * A202138 A202139 A202140 KEYWORD nonn AUTHOR Zak Seidov, Dec 15 2011 STATUS approved

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Last modified December 8 07:38 EST 2022. Contains 358691 sequences. (Running on oeis4.)