

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
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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



