A045752 4n-1 is composite.

4, 7, 9, 10, 13, 14, 16, 19, 22, 23, 24, 25, 28, 29, 30, 31, 34, 36, 37, 39, 40, 43, 44, 46, 47, 49, 51, 52, 54, 55, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 86, 88, 89, 91, 93, 94, 97, 98, 99, 100

Offset: 1

Author

Felice Russo

Keywords

• nonn
• easy

Comments

Apparently the same as "numbers k that can be written as 4xy + x - y for x>0,y>0". - Ron R Spencer, Jul 28 2016

From Wolfdieter Lang, Aug 30 2016: (Start)

Proof: If the 3 (mod 4) number 4*k-1 is composite it can be written as a product of a number a == 3 (mod 4) and powers of numbers 1 (mod 4), that is as a product of

a = 4*x-1 and b = 4*y+1. Then 4*k-1 = (4*x-1)*(4*y+1) or k = 4*x*y + x - y. And conversely, if k = 4*x*y + x - y then 4*k-1 = (4*x-1)*(4*y+1), that is composite.

The example of Vincenzo Librandi below is equivalent to "numbers m that can be written as 4*H*K + 3*H + K +1 for H>0, K>0" (consider h, k of opposite parity, which is necessary to have even 2*h*k + k + h + 1. W.l.o.g. take h = 2*H and k = 2*K+1). Then 4*m - 1 = (4*K+3)*(4*H+1). This is equivalent to Ron R Spencer's statement with K=x-1, H=y. (End)

Examples

			7 belongs to the sequence because 7*4-1=27 is not a prime.
Distribution of the positive terms in the following triangular array:
*;
4,*;
*,9,*;
7,*,16,*;
*,14,*,25,*;
10,*,23,*,36,*; etc.
where * marks the non-integer values of (2*h*k + k + h + 1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jul 29 2016
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A005099.

Programs

Magma

[n: n in [1..120] |not IsPrime(4*n-1)]; // Vincenzo Librandi, Jul 29 2016

Maple

remove(t -> isprime(4*t-1), [$1..1000]); # Robert Israel, Jul 29 2016

Mathematica

Select[Range@ 100, CompositeQ[4 # - 1] &] (* Michael De Vlieger, Jul 28 2016 *)

PARI

isok(n) = ! isprime(4*n-1); \\ Michel Marcus, Sep 28 2013