%I #96 May 16 2023 07:05:49

%S 2047,3277,4033,4681,8321,15841,29341,42799,49141,52633,65281,74665,

%T 80581,85489,88357,90751,104653,130561,196093,220729,233017,252601,

%U 253241,256999,271951,280601,314821,357761,390937,458989,476971,486737

%N Strong pseudoprimes to base 2.

%C The number 2^k-1 is in the sequence iff k is in A054723 or in A001567. - _Thomas Ordowski_, Sep 02 2016

%C The number (2^k+1)/3 is in the sequence iff k is in A127956. - _Davide Rotondo_, Aug 13 2021

%D R. K. Guy, Unsolved Problems Theory Numbers, A12.

%D P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 95.

%H T. D. Noe, <a href="/A001262/b001262.txt">Table of n, a(n) for n = 1..10000</a> (using data from A001567)

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.10, pp. 786-792.

%H Chris Caldwell, <a href="https://t5k.org/glossary/xpage/StrongPRP.html">Strong probable prime</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongPseudoprime.html">Strong Pseudoprime</a>.

%H OEIS Wiki, <a href="http://oeis.org/wiki/Strong_pseudoprimes">Strong Pseudoprime</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Strong_pseudoprime">Strong pseudoprime</a>.

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%e From _Michael B. Porter_, Sep 04 2016: (Start)

%e For k = 577, k-1 = 576 = 9*2^6. Since 2^(9*2^3) = 2^72 == -1 (mod 577), 577 passes the primality test, but since it is actually prime, it is not in the sequence.

%e For k = 3277, k-1 = 3276 = 819*2^2, and 2^(819*2) == -1 (mod 3277), so k passes the primality test, and k = 3277 = 29*113 is composite, so 3277 is in the sequence. (End)

%p A007814 := proc(n) padic[ordp](n,2) ; end proc:

%p isStrongPsp := proc(n,b) local d,s,r; if type(n,'even') or n<=1 then return false; elif isprime(n) then return false; else s := A007814(n-1) ; d := (n-1)/2^s ; if modp(b &^ d,n) = 1 then return true; else for r from 0 to s-1 do if modp(b &^ d,n) = n-1 then return true; end if; d := 2*d ; end do: return false; end if; end if; end proc:

%p isA001262 := proc(n) isStrongPsp(n,2) ; end proc:

%p for n from 1 by 2 do if isA001262(n) then print(n); end if; end do:

%p # _R. J. Mathar_, Apr 05 2011

%t 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, n] == n-1, Return[True]]; d = 2*d, {s}]]); lst = {}; k = 3; While[k < 500000, If[sppQ[k, 2], Print[k]; AppendTo[lst, k]]; k += 2]; lst (* _Jean-François Alcover_, Oct 20 2011, after _R. J. Mathar_ *)

%o (PARI)

%o isStrongPsp(n,b)={

%o my(s,d,r,bm) ;

%o if( (n% 2) ==0 || n <=1, return(0) ;) ;

%o if(isprime(n), return(0) ;) ;

%o s = valuation(n-1,2) ;

%o d = (n-1)/2^s ;

%o bm = Mod(b,n)^d ;

%o if ( bm == Mod(1,n), return(1) ;) ;

%o for(r=0,s-1,

%o bm = Mod(b,n)^d ;

%o if ( bm == Mod(-1,n),

%o return(1) ;

%o ) ;

%o d *= 2;

%o ) ;

%o return(0);

%o }

%o isA001262(n)={

%o isStrongPsp(n,2)

%o }

%o {

%o for(n=1,10000000000,

%o if(isA001262(n),

%o print(n)

%o ) ;

%o ) ;

%o } \\ _R. J. Mathar_, Mar 07 2012

%o (PARI) is_A001262(n,a=2)={ (bittest(n,0) && !isprime(n) && n>8) || return; my(s=valuation(n-1,2)); if(1==a=Mod(a,n)^(n>>s),return(1)); while(a!=-1 && s--, a=a^2); a==-1} \\ _M. F. Hasler_, Aug 16 2012

%Y Cf. A001567 (pseudoprimes to base 2), A020229 (strong pseudoprimes to base 3), A020231 (base 5), A020233 (base 7).

%Y Cf. A072276 (SPP to base 2 and 3), A215568 (SPP to base 2 and 5), A056915 (SPP to base 2,3 and 5), A074773 (SPP to base 2,3,5 and 7).

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_, Aug 15 1996