cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A072276 Strong pseudoprimes to bases 2 and 3.

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%I A072276 #46 Feb 16 2025 08:32:46
%S A072276 1373653,1530787,1987021,2284453,3116107,5173601,6787327,11541307,
%T A072276 13694761,15978007,16070429,16879501,25326001,27509653,27664033,
%U A072276 28527049,54029741,61832377,66096253,74927161,80375707,101649241
%N A072276 Strong pseudoprimes to bases 2 and 3.
%C A072276 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).
%C A072276 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
%H A072276 Don Reble, <a href="/A072276/b072276.txt">Table of n, a(n) for n = 1..10000</a>
%H A072276 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.10, pp. 786-792
%H A072276 D. Bleichenbacher, <a href="http://www.bell-labs.com/user/bleichen/diss/thesis.html">Thesis and strong pseudoprimes to 2 and 3 up to 10^16</a>
%H A072276 Washington Bomfim, <a href="http://dl.dropbox.com/u/79527920/bF23.txt">Table of n, a(n) for n=1..1499371</a> [a large file]
%H A072276 Jan Feitsma and William Galway, <a href="http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html">Tables of pseudoprimes and related data</a>
%H A072276 A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://www.cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, section 4.2.3, Miller-Rabin test.
%H A072276 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html">Rabin-Miller test</a>
%H A072276 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%t A072276 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_ *)
%Y A072276 Cf. A001262, A006945, A014233, A020229.
%K A072276 nonn
%O A072276 1,1
%A A072276 _Francois R. Grieu_, Jul 09 2002