

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



MAPLE

filter:= n > not issqr(24*n+1) and not isprime(24*n+1):


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).


KEYWORD

nonn


AUTHOR



STATUS

approved



