A020229 -id:A020229 - cp's OEIS frontend

A056915 Strong pseudoprimes to bases 2, 3 and 5, i.e., intersection of A001262, A020229, and A020231.

25326001, 161304001, 960946321, 1157839381, 3215031751, 3697278427, 5764643587, 6770862367, 14386156093, 15579919981, 18459366157, 19887974881, 21276028621, 27716349961, 29118033181, 37131467521, 41752650241, 42550716781, 43536545821

Offset: 1

Rick L. Shepherd, Feb 12 2002

These first 13 numbers are the only ones less than 25*10^9 which are simultaneously strong pseudoprimes to bases 2, 3 and 5. Taken from the same table - which indicates (only) whether they are also strong pseudoprime (spsp) or pseudoprime (psp) to base 7, 11 and/or 13: 161304001 is spsp to 11; 3215031751 is spsp to base 7 and is psp to both bases 11 and 13; 5764643587 is spsp to base 13; 14386156093 is psp to bases 7, 11 and 13. 15579919981 is psp to base 7 and spsp to base 11; 19887974881 is psp to base 7; and 21276028621 is psp to bases 11 and 13.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, NY, 1991, pp. 82-83.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Pomerance, C., Selfridge, J.L. and Wagstaff, Jr., S.S. The pseudoprimes to 25*10^9, Mathematics of Computation 35, 1980, pp. 1003-1026.
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A072276, A001262, A020229, A020231, superset of A074773.

B-file and more terms from Charles R Greathouse IV, Aug 14 2010

A001262 Strong pseudoprimes to base 2.

Original entry on oeis.org

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

N. J. A. Sloane

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

			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)

R. K. Guy, Unsolved Problems Theory Numbers, A12.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 95.

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

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).

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

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 *)

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

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

More terms from David W. Wilson, Aug 15 1996

A020231 Strong pseudoprimes to base 5.

Original entry on oeis.org

781, 1541, 5461, 5611, 7813, 13021, 14981, 15751, 24211, 25351, 29539, 38081, 40501, 44801, 53971, 79381, 100651, 102311, 104721, 112141, 121463, 133141, 141361, 146611, 195313, 211951, 216457, 222301, 251521, 289081, 290629, 298271, 315121

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..715 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A005936, A001262 (base-2 SPP), A020229 (base-3 SPP), A215568 (SPP to bases 2 & 5), A215566 (SPP to bases 3 & 5), A056915 (SPP to bases 2, 3 & 5), A074773 (SPP to bases 2, 3, 5 & 7).

nmax = 400000; 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}]]); A020231 = {}; n = 1; While[n < nmax, n = n+2; If[sppQ[n, 5] == True, Print[n]; AppendTo[A020231, n]]]; A020231 (* Jean-François Alcover, Oct 20 2011, after R. J. Mathar *)

A020233 Strong pseudoprimes to base 7.

Original entry on oeis.org

25, 325, 703, 2101, 2353, 4525, 11041, 14089, 20197, 29857, 29891, 39331, 49241, 58825, 64681, 76627, 78937, 79381, 87673, 88399, 88831, 102943, 109061, 137257, 144901, 149171, 173951, 178709, 188191, 197633, 219781, 227767, 231793, 245281

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A005938, A001262, A020229, A020230, A020231, A020232.

A074773 Strong pseudoprimes to bases 2, 3, 5 and 7.

Original entry on oeis.org

3215031751, 118670087467, 307768373641, 315962312077, 354864744877, 457453568161, 528929554561, 546348519181, 602248359169, 1362242655901, 1871186716981, 2152302898747, 2273312197621, 2366338900801, 3343433905957, 3461715915661, 3474749660383, 3477707481751, 4341937413061, 4777422165601, 5537838510751

Offset: 1

Don Reble, Sep 07 2002

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Washington Bomfim, Table with all 16757 terms up to 2^64
G. Jaeschke, On strong pseudoprimes to several bases, Mathematics of Computation, 61 (1993), 915-926.
Eric Weisstein's World of Mathematics, Miller's Primality Test
Index entries for sequences related to pseudoprimes

Cf. A001262, A020229, A020231, A020233, A056915.

sprp(n,b)=my(s=valuation(n-1,2),d=Mod(b,n)^(n>>s)); if(d==1, return(1)); for(i=1,s-1, if(d==-1, return(1)); d=d^2;); d==-1
is(n)=sprp(n,2) && sprp(n,3) && sprp(n,5) && sprp(n,7) && !isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

b-file, link, and editing from Charles R Greathouse IV, Aug 14 2010

A020230 Strong pseudoprimes to base 4.

Original entry on oeis.org

341, 1387, 2047, 3277, 4033, 4371, 4681, 5461, 8321, 8911, 10261, 13747, 14491, 15709, 15841, 19951, 29341, 31621, 42799, 49141, 49981, 52633, 60787, 65077, 65281, 74665, 80581, 83333, 85489, 88357, 90751, 104653, 123251, 129921, 130561, 137149

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..229 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A020136 (base 4), A001262 (base 2), A020229 (base 3), A020231 (base 5), A020232 (base 6), A020233 (base 7), A020234 (base 8), A020235 (base 9), A020236 (base 10), A020237 (base 11), A020238 (base 12).

A020232 Strong pseudoprimes to base 6.

Original entry on oeis.org

217, 481, 1111, 1261, 2701, 3589, 5713, 6533, 11041, 14701, 20017, 29341, 34441, 39493, 43621, 46657, 46873, 49141, 49661, 58969, 74023, 74563, 76921, 83333, 87061, 92053, 94657, 94697, 97751, 97921, 109061, 115921, 125563, 128627, 151387, 173377

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..210 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A005937, A001262, A020229, A020230, A020231.

A020234 Strong pseudoprimes to base 8.

Original entry on oeis.org

9, 65, 481, 511, 1417, 2047, 2501, 3277, 3641, 4033, 4097, 4681, 8321, 11041, 15841, 16589, 19561, 24311, 24929, 29341, 41441, 42799, 45761, 49141, 52429, 52633, 54161, 55969, 56033, 59291, 61337, 65281, 66197, 74023, 74665, 77161, 80581, 85489, 87061

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..237 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A020137, A001262, A020229, A020230, A020231, A020232, A020233.

A020235 Strong pseudoprimes to base 9.

Original entry on oeis.org

91, 121, 671, 703, 1541, 1729, 1891, 2821, 3281, 3367, 3751, 5551, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 16471, 16531, 18721, 19345, 23521, 24661, 24727, 28009, 29341, 30857, 31621, 32791, 38503, 44287, 46999, 47197, 49051, 49141, 53131

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..369 from R. J. Mathar)
Eric Weisstein's World of Mathematics, Strong Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A020138, A020229 (base 3), A020307 (base 81).

A071294 Number of witnesses for strong pseudoprimality of 2n+1, i.e., number of bases b, 1 <= b <= 2n, in which 2n+1 is a strong pseudoprime.

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 2, 2, 58, 60, 2, 6, 66, 2, 70, 72, 2, 2, 78, 2, 82, 6, 2, 88, 18, 2, 2, 96, 2, 100, 102, 2, 106, 108, 2, 112, 2, 2, 2, 10, 2, 4, 126, 2, 130, 18, 2, 136, 138, 2, 2, 6, 2, 148, 150, 2, 2, 156, 2, 2

Offset: 1

J.-F. Guiffes (guiffes.jean-francois(AT)wanadoo.fr), Jun 11 2002

nonn

Number of integers b, 1 <= b <= 2n, such that if 2n = 2^k*m with odd m, then the sequence (b^m, b^(2*m), ..., b^(2^k*m)) modulo 2n+1 satisfies the Rabin-Miller test.
Comments from R. J. Mathar, Jul 03 2012 (Start)
The subsequence related to composite 2n+1 is characterized with records in A195328 and associated 2n+1 tabulated in A141768.
Let N = 2n+1 = product_{i=1..s} p_i^r_i be the prime factorization of the odd 2n+1. Related odd parts q and q_i are defined by N-1=2^k*q and p_i-1 = 2^(k_i)*q_i, with sorting such that k_1 <= k_2 <=k_3... Then a(n) = (1+sum_{j=0..k1-1} 2^(j*s)) *product_{i=1..s} gcd(q,qi).
Reduces to A006093 if 2n+1 is prime.
This might be correlated with 2*A195508(n). (End)

Paulo Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer-Verlag, New York, 2004, p. 98.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.
Wikipedia, Strong pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A000265, A001221, A006093, A007814, A063994, A141768, A195328.
Cf. also A001262, A020229, A020231, A020233.
Different from A060684.

rabinmiller := proc(n,a); k := 0; mu := n-1; while irem(mu,2)=0 do k := k+1; mu := mu/2 od; G := a&^mu mod(n); h := 0; if G=1 then RETURN(1) else while h1 do h := h+1; G := G&^2 mod n; od; if h n-1 then RETURN(0) else RETURN(1) fi; if G=1 then RETURN(1); fi; fi; end; compte := proc(n) local l; RETURN(sum('rabinmiller(2*n+1,l)','l'=1..2*n)); end;
Maple code from R. J. Mathar, Jul 03 2012 (Start)
A000265 := proc(n)
     n/2^padic[ordp](n,2) ;
end proc:
a := proc(n)
     q := A000265(n-1) ;
     B := 1;
     s := 0 ;
     k1 := 10000000000000 ;
     for pf in ifactors(n)[2] do
         pi := op(1,pf) ;
         qi := A000265(pi-1) ;
         ki := ilog2((pi-1)/qi) ;
         k1 := min(k1,ki) ;
         B := B*igcd(q,qi) ;
         s := s+1 ;
     end do:
     1+add(2^(j*s),j=0..k1-1) ;
     return B*% ;
end proc:
seq(a(2*n+1),n=1..60) ;

o[n_] := (n-1)/2^IntegerExponent[n-1, 2]; a[n_?PrimeQ] := n-1; a[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, om = Length[p]; Product[GCD[o[n], o[p[[k]]]], {k, 1, om}] * (1 + (2^(om * Min[IntegerExponent[#, 2]& /@ (p - 1)]) - 1)/(2^om - 1))]; Table[a[n], {n, 3, 121, 2}] (* Amiram Eldar, Nov 08 2019 *)

For k = 2*n+1, a(k) = k - 1 if k is prime, otherwise, a(k) = (1 + 2^(omega(k)*nu(k)) - 1)/(2^omega(k)-1)) * Product_{p|k} gcd(od(k-1), od(p-1)), where omega(m) is the number of distinct prime factors of m (A001221), od(m) is the largest odd divisor of m (A000265) and nu(m) = min_{p|m} A007814(p-1). - Amiram Eldar, Nov 08 2019

Edited by Max Alekseyev, Sep 20 2018

Edited by N. J. A. Sloane, Nov 15 2019, merging R. J. Mathar's A182291 with this entry.

cp's OEIS Frontend

A056915 Strong pseudoprimes to bases 2, 3 and 5, i.e., intersection of A001262, A020229, and A020231.

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A001262 Strong pseudoprimes to base 2.

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A020231 Strong pseudoprimes to base 5.

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Mathematica

A020233 Strong pseudoprimes to base 7.

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A074773 Strong pseudoprimes to bases 2, 3, 5 and 7.

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A020230 Strong pseudoprimes to base 4.

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A020232 Strong pseudoprimes to base 6.

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A020234 Strong pseudoprimes to base 8.

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A020235 Strong pseudoprimes to base 9.

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A071294 Number of witnesses for strong pseudoprimality of 2n+1, i.e., number of bases b, 1 <= b <= 2n, in which 2n+1 is a strong pseudoprime.

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