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A001262
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Strong pseudoprimes to base 2.
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71
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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
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OFFSET
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1,1
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COMMENTS
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The number 2^n-1 is in the sequence iff n is in A054723 or in A001567. - Thomas Ordowski, Sep 02 2016
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REFERENCES
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R. K. Guy, Unsolved Problems Theory Numbers, A12.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 95.
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LINKS
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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, Pseudoprime
Eric Weisstein's World of Mathematics, Strong Pseudoprime
OEIS Wiki, Strong Pseudoprime
Wikipedia, Strong pseudoprime
Index entries for sequences related to pseudoprimes
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EXAMPLE
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For n = 577, n-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 n = 3277, n-1 = 3276 = 819*2^2, and 2^(819*2) == -1 (mod 3277), so n passes the primality test, and n = 3277 = 29*113 is composite, so 3277 is in the sequence. - Michael B. Porter, Sep 04 2016
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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 A062568 A180065
Adjacent sequences: A001259 A001260 A001261 * A001263 A001264 A001265
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from David W. Wilson, Aug 15 1996
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STATUS
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approved
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