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%I #38 Sep 08 2022 08:44:56
%S 4,7,9,10,13,14,16,19,22,23,24,25,28,29,30,31,34,36,37,39,40,43,44,46,
%T 47,49,51,52,54,55,58,59,61,62,64,65,67,69,70,72,73,74,75,76,79,80,81,
%U 82,84,85,86,88,89,91,93,94,97,98,99,100
%N 4n-1 is composite.
%C 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
%C From _Wolfdieter Lang_, Aug 30 2016: (Start)
%C 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
%C 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.
%C 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)
%H Robert Israel, <a href="/A045752/b045752.txt">Table of n, a(n) for n = 1..10000</a>
%e 7 belongs to the sequence because 7*4-1=27 is not a prime.
%e Distribution of the positive terms in the following triangular array:
%e *;
%e 4,*;
%e *,9,*;
%e 7,*,16,*;
%e *,14,*,25,*;
%e 10,*,23,*,36,*; etc.
%e where * marks the non-integer values of (2*h*k + k + h + 1)/2 with h >= k >= 1. - _Vincenzo Librandi_, Jul 29 2016
%p remove(t -> isprime(4*t-1), [$1..1000]); # _Robert Israel_, Jul 29 2016
%t Select[Range@ 100, CompositeQ[4 # - 1] &] (* _Michael De Vlieger_, Jul 28 2016 *)
%o (PARI) isok(n) = ! isprime(4*n-1); \\ _Michel Marcus_, Sep 28 2013
%o (Magma) [n: n in [1..120] |not IsPrime(4*n-1)]; // _Vincenzo Librandi_, Jul 29 2016
%Y Complement of A005099.
%K nonn,easy
%O 1,1
%A _Felice Russo_
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