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A072276 Strong pseudoprimes to bases 2 and 3. 7
1373653, 1530787, 1987021, 2284453, 3116107, 5173601, 6787327, 11541307, 13694761, 15978007, 16070429, 16879501, 25326001, 27509653, 27664033, 28527049, 54029741, 61832377, 66096253, 74927161, 80375707, 101649241 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Oct 20 2011, after R. J. Mathar *)

CROSSREFS

Cf. A001262, A020229, A006945, A014233.

Sequence in context: A235855 A191820 A074999 * A114657 A250058 A205205

Adjacent sequences:  A072273 A072274 A072275 * A072277 A072278 A072279

KEYWORD

nonn

AUTHOR

Francois Grieu (fgrieu(AT)micronet.fr), Jul 09 2002

STATUS

approved

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Last modified December 19 22:28 EST 2014. Contains 252240 sequences.