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
Examples
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)
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
Crossrefs
Programs
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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
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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 *)
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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
Extensions
More terms from David W. Wilson, Aug 15 1996
Comments