A072276 Strong pseudoprimes to bases 2 and 3.

1373653, 1530787, 1987021, 2284453, 3116107, 5173601, 6787327, 11541307, 13694761, 15978007, 16070429, 16879501, 25326001, 27509653, 27664033, 28527049, 54029741, 61832377, 66096253, 74927161, 80375707, 101649241

Offset

1,1

Comments

Composites that pass the Miller-Rabin test for bases 2 and 3. The intersection of A001262 (strong pseudoprimes to base 2) and A020229 (strong pseudoprimes to base 3).

The Washington Bomfim link references a table with all terms up to 2^64. Data from Jan Feitsma and William Galway, see link below, permitted an easy determination of these terms. I tested the Mathematica function PrimeQ[n] with those numbers to verify that it is correct for all n < 2^64. - Washington Bomfim, May 13 2012

Links

Don Reble, Table of n, a(n) for n = 1..10000

Joerg Arndt, Matters Computational (The Fxtbook), section 39.10, pp. 786-792

D. Bleichenbacher, Thesis and strong pseudoprimes to 2 and 3 up to 10^16

Washington Bomfim, Table of n, a(n) for n=1..1499371 [a large file]

Jan Feitsma and William Galway, Tables of pseudoprimes and related data

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, section 4.2.3, Miller-Rabin test.

Eric Weisstein's World of Mathematics, Rabin-Miller test

Index entries for sequences related to pseudoprimes

Mathematica

nmax = 10^8; sppQ[n_?EvenQ, _] := False; sppQ[n_?PrimeQ, _] := False; sppQ[n_, b_] := (s = IntegerExponent[n - 1, 2]; d = (n - 1)/2^s; If[ PowerMod[b, d, n] == 1, Return[True], Do[ If[ PowerMod[b, d*2^r, n] == n-1, Return[True]], {r, 0,  s-1}]]); A072276 = {}; n = 1; While[n < nmax, n = n+2; If[sppQ[n, 2] && sppQ[n, 3] , Print[n]; AppendTo[ A072276, n]]]; A072276 (* Jean-François Alcover, Oct 20 2011, after R. J. Mathar *)

Crossrefs

Cf. A001262, A006945, A014233, A020229.

Sequence in context: A191820 A074999 A347279 * A114657 A250058 A205205

Adjacent sequences: A072273 A072274 A072275 * A072277 A072278 A072279

Keyword

nonn

Author

Francois R. Grieu, Jul 09 2002

Status

approved