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

%I A216170 #14 Feb 16 2025 08:33:18
%S A216170 341,645,2465,2821,4033,5461,8321,15841,25761,31621,68101,83333,
%T A216170 162401,219781,282133,348161,530881,587861,653333,710533,722261,
%U A216170 997633,1053761,1082401,1193221,1246785,1333333,1357441,1398101,1489665,1584133,1690501,1735841
%N A216170 Fermat pseudoprimes to base 2 of the form (n^2 + 2*n)/3.
%C A216170 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.
%C A216170 The formula can be generalized this way: Fermat pseudoprimes to base 2 of the form (n^m + m*n)/(m+1).
%C A216170 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.
%C A216170 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).
%H A216170 Charles R Greathouse IV, <a href="/A216170/b216170.txt">Table of n, a(n) for n = 1..10000</a>
%H A216170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%t A216170 t = Select[Table[n (n + 2)/3, {n, 4, 10000}], IntegerQ]; Select[t, PowerMod[2, # - 1, #] == 1 &] (* _T. D. Noe_, Sep 03 2012 *)
%o A216170 (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
%Y A216170 Cf. A001567, A210454 (subsequence).
%K A216170 nonn
%O A216170 1,1
%A A216170 _Marius Coman_, Sep 03 2012