A215326 - OEIS

%I #12 Dec 07 2014 21:24:44

%S 341,645,1105,1387,2047,2465,2821,3277,4033,5461,6601,7957,8321,11305,

%T 13747,15841,16705,19951,23001,25761,30889,31417,31621,39865,41665,

%U 49981,65077,68101,74665,83333,83665,85489,88357,90751,107185,137149,158369

%N Fermat pseudoprimes to base 2 of the form m*n^2 + (11*m - 23)*n + 19*m - 49, where m, n >= 0.

%C Fermat pseudoprimes to base 2 are also called Poulet numbers.

%C The solutions (m,n) for the Poulet numbers from the sequence above are (3,9); (3,13); (4,14); (4,16); (9,11) and (4,20); (3,27); (3,29); (4,26); (3,35); (290,0) and (3,41); (350,0) and (4,38); (259,1); (3,51); (367,1); (4,56); (94,8) and (3,71); (4,62); (329,3) and (4,68); (379,3); (3,91); (182,8); (319,5) and (4,86); (3,101); (888,2); (928,2) and (66,20); (43,29); (659,5); (3,149); (438,8) and (4,134); (3,165); (4406,0) and (4,142); (4502,0); (4,146); (4,148); (2384,2) and (38,48); (1387,5); (5111,1).

%C Note: for n = -2 the formula becomes (m - 3), and for n = -9 it becomes (m + 158), so all the Poulet numbers have at least these integer solutions to this formula.

%C Another interesting observation about the integer solutions: for n = -1 the formula becomes (9*m - 26), and 37 of the first 100 Poulet numbers can be written this way! This means that, for more than a third of Poulet numbers P checked, it is true that (P + 8) is divisible by 9 (for comparison, this relation is true for just 14 of the first 100 primes).

%H Charles R Greathouse IV, <a href="/A215326/b215326.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>

%e For m = 3 the formula is 3*n^2 + 10*n + 8, giving the following Poulet numbers: 341, 645, 2465, 2821, 4033, 5461, 8321, 15841, 25761, 31621, 68101, 83333, etc.

%e For m = 4 the formula is 4*n^2 + 21*n + 27, giving the following Poulet numbers: 1105, 1387, 2047, 3277, 6601, 13747, 16705, 19951, 31417, 83665, 88357, 90751, etc.

%o (PARI) is(n)=if(Mod(2,n)^(n-1)!=1||isprime(n), return(0)); for(k=0,sqrtint(n+66)+6, if(Mod(n,k^2+11*k+19)+23*k+49==0, return(1))); 0 \\ _Charles R Greathouse IV_, Dec 07 2014

%Y Cf. A001567.

%K nonn

%O 1,1

%A _Marius Coman_, Aug 08 2012