A001262 Strong pseudoprimes to base 2.

2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751, 104653, 130561, 196093, 220729, 233017, 252601, 253241, 256999, 271951, 280601, 314821, 357761, 390937, 458989, 476971, 486737

Offset

1,1

Comments

The number 2^k-1 is in the sequence iff k is in A054723 or in A001567. - Thomas Ordowski, Sep 02 2016

The number (2^k+1)/3 is in the sequence iff k is in A127956. - Davide Rotondo, Aug 13 2021

References

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

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

Links

T. D. Noe, Table of n, a(n) for n = 1..10000 (using data from A001567)

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

Chris Caldwell, Strong probable prime.

Eric Weisstein's World of Mathematics, Strong Pseudoprime.

OEIS Wiki, Strong Pseudoprime.

Wikipedia, Strong pseudoprime.

Index entries for sequences related to pseudoprimes

Example

From Michael B. Porter, Sep 04 2016: (Start)
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.
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)

Maple

A007814 := proc(n) padic[ordp](n, 2) ; end proc:
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:
isA001262 := proc(n) isStrongPsp(n, 2) ; end proc:
for n from 1 by 2 do if isA001262(n) then print(n); end if; end do:
# R. J. Mathar, Apr 05 2011

Mathematica

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 *)

Prog

(PARI)
isStrongPsp(n, b)={
        my(s, d, r, bm) ;
        if( (n% 2) ==0 || n <=1, return(0) ; ) ;
        if(isprime(n), return(0) ; ) ;
        s = valuation(n-1, 2) ;
        d = (n-1)/2^s ;
        bm = Mod(b, n)^d ;
        if ( bm == Mod(1, n), return(1) ; ) ;
        for(r=0, s-1,
                bm = Mod(b, n)^d ;
                if ( bm == Mod(-1, n),
                        return(1) ;
                ) ;
                d *= 2;
        ) ;
        return(0);
}
isA001262(n)={
        isStrongPsp(n, 2)
}
{
for(n=1, 10000000000,
    if(isA001262(n),
        print(n)
    ) ;
) ;
} \\ R. J. Mathar, Mar 07 2012
(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

Crossrefs

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

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).

Sequence in context: A241039 A278353 A038462 * A141232 A367230 A361256

Adjacent sequences: A001259 A001260 A001261 * A001263 A001264 A001265

Keyword

nonn,nice,changed

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Aug 15 1996

Status

approved