A306247 - OEIS

%I #25 Aug 09 2019 16:52:20

%S 1,2,3,6,9,14,15,19,21,26,29,30,34,36,40,48,49,51,54,61,63,64,69,74,

%T 75,79,82,84,86,89,90,95,96,99,103,106,110,111,112,114,119,120,135,

%U 139,141,146,147,149,151,152,153,154,156,161,166,169,173,174,179,180,184,186,187,190,194

%N Numbers k such that 2k - p is not a prime where p is any prime divisor of 4k^2 - 1.

%C Primes in a(n): 2, 3, 19, 29, 61, 79, 89, 103, 139, 149, 151, 173, 179, ...

%F A306261(a(n)) > 1 for n >= 4.

%e 1 is a term because 4*1^2 - 1 = 3 and 2*1 - 3 = -1 (nonprime);

%e 2 is a term because 4*2^2 - 1 = 15 and 2*2 - 15 = -11 (nonprime);

%e 3 is a term because 4*3^2 - 1 = 35 and 2*3 - 35 = -29 (nonprime);

%e 6 is a term because 4*6^2 - 1 = 143 = 11*13 and 2*6 - 11 = 1 (nonprime), 2*6 - 13 = -1 (nonprime);

%e 9 is a term because 4*9^2 - 1 = 323 = 17*19 and 2*9 - 17 = 1 (nonprime), 2*9 - 19 = -1 (nonprime).

%p filter:= proc(n) andmap(`not` @ isprime, map(p -> 2*n-p, numtheory:-factorset(4*n^2-1))) end proc:

%p select(filter, [$1..300]); # _Robert Israel_, Jan 31 2019

%t Select[Range@ 200, AllTrue[2 # - FactorInteger[4 #^2 - 1][[All, 1]], ! PrimeQ@ # &] &] (* _Michael De Vlieger_, Feb 03 2019 *)

%o (PARI) isok(k) = {my(pf = factor(4*k^2-1)[,1]); #select(x->isprime(2*k-x), pf) == 0;} \\ _Michel Marcus_, Mar 02 2019

%Y Includes A040040.

%Y Cf. A306261.

%K nonn,easy

%O 1,2

%A _Juri-Stepan Gerasimov_, Jan 31 2019