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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
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):
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
KEYWORD
nonn
AUTHOR
Zak Seidov, Dec 15 2011
STATUS
approved