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

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

Offset: 1

Author

Zak Seidov, Dec 15 2011

Keywords

• nonn

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

Crossrefs

Cf. A089237 (list of primes and squares), A089229 (neither primes nor squares).

Programs

Magma

[n: n in [1..200] | not IsSquare(24*n+1) and not IsPrime(24*n+1)]; // Vincenzo Librandi, Jan 26 2018

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 *)

PARI

for(n=1,200,m=24*n+1;if(isprime(m)+issquare(m),,print1(n",")))