A216170 Fermat pseudoprimes to base 2 of the form (n^2 + 2*n)/3.

341, 645, 2465, 2821, 4033, 5461, 8321, 15841, 25761, 31621, 68101, 83333, 162401, 219781, 282133, 348161, 530881, 587861, 653333, 710533, 722261, 997633, 1053761, 1082401, 1193221, 1246785, 1333333, 1357441, 1398101, 1489665, 1584133, 1690501, 1735841

Offset: 1

Author

Marius Coman, Sep 03 2012

Keywords

• nonn

Comments

The corresponding values of n: 31, 43, 85, 91, 109, 127, 157, 217, 277, 307, 451, 499, 697, 811, 919, 1021, 1261, 1327, 1399, 1459, 1471, 1729, 1777, 1801, 1891, 1933, 1999, 2017, 2047, 2113, 2177, 2251.

The formula can be generalized this way: Fermat pseudoprimes to base 2 of the form (n^m + m*n)/(m+1).

For m = 3, the formula becomes (n^3 + 3*n)/4, from which the Poulet numbers 341, 1729, 188461, 228241, and 1082809 (for n = 11, 19, 91, 97, and 163, respectively) were obtained.

Conjecture: For any m natural, m > 1, there exists a series with infinitely many Fermat pseudoprimes to base 2, P, formed this way: P = (n^m + m*n)/(m+1).

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Poulet Number

Crossrefs

Cf. A001567, A210454 (subsequence).

Programs

Mathematica

t = Select[Table[n (n + 2)/3, {n, 4, 10000}], IntegerQ]; Select[t, PowerMod[2, # - 1, #] == 1 &] (* T. D. Noe, Sep 03 2012 *)

PARI

list(lim)=my(v=List(),t); lim\=1; forstep(n=31,sqrtint(3*lim+1)-1,[2,1], t=t=n*(n+2)/3; if(Mod(2,t)^t==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017