

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
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OFFSET

1,1


COMMENTS

Composites that pass the MillerRabin 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. 786792
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, MillerRabin test
Eric Weisstein's World of Mathematics, RabinMiller 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] == n1, Return[True]], {r, 0, s1}]]); A072276 = {}; n = 1; While[n < nmax, n = n+2; If[sppQ[n, 2] && sppQ[n, 3] , Print[n]; AppendTo[ A072276, n]]]; A072276 (* JeanFrançois 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



