A181780 Numbers n which are Fermat pseudoprimes to some base b, 2 <= b <= n-2.

15, 21, 25, 28, 33, 35, 39, 45, 49, 51, 52, 55, 57, 63, 65, 66, 69, 70, 75, 76, 77, 85, 87, 91, 93, 95, 99, 105, 111, 112, 115, 117, 119, 121, 123, 124, 125, 129, 130, 133, 135, 141, 143, 145, 147, 148, 153, 154, 155, 159, 161, 165, 169, 171, 172, 175, 176

Offset: 1

Author

Karsten Meyer, Nov 12 2010

Keywords

• nonn

Comments

A nonprime number n is a Fermat pseudoprime to base b if b^(n-1) = 1 (mod n).

It appears that these n are pseudoprimes for an even number of bases. When n is the product of two distinct primes, it appears that there are exactly two such bases x and y with x + y = n. See A211455, A211456, and A211457. - T. D. Noe, Apr 12 2012

Examples

			15 is Fermat pseudoprime to base 4 and 11, so it is a Fermat pseudoprime.
		

Links

• Karsten Meyer and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 5978 terms from Karsten Meyer)
• Karsten Meyer, Tabelle Pseudoprimzahlen (15-4999)
• Karsten Meyer, Rexx program for this sequence
• Eric Weisstein's World of Mathematics, Fermat Pseudoprime
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A039769, A181781, A211455, A211456, A211457, A211458, A227180, A280199.

Even terms give A039772. - Thomas Ordowski, Dec 28 2016

Programs

Mathematica

t = {}; Do[s = Select[Range[2, n-2], PowerMod[#, n-1, n] == 1 &]; If[s != {}, AppendTo[t, n]], {n, Select[Range[213], ! PrimeQ[#] &]}]; t (* T. D. Noe, Nov 07 2011 *)
(* The following program is much faster than the one above. See A227180 for indications of a proof of this assertion. *) Select[Range[213], ! IntegerQ[Log[3, #]] && ! PrimeQ[#] && GCD[# - 1, EulerPhi[#]] > 1 &] (* Emmanuel Vantieghem, Jul 06 2013 *)

PARI

fsp(n)=
{ /* whether n is Fermat pseudoprime to any base a where 2<=a<=n-2 */
    for (a=2,n-2,
        if ( gcd(a,n)!=1, next() );
        if ( (Mod(a,n))^(n-1)==+1, return(1) )
    );
    return(0);
}
for(n=3,300, if(isprime(n),next());  if ( fsp(n) , print1(n,", ") ); );
\\ Joerg Arndt, Jan 08 2011

PARI

is(n)=if(isprime(n), return(0)); my(f=factor(n)[,1]); prod(i=1, #f, gcd(f[i]-1, n-1)) > 2 \\ Charles R Greathouse IV, Dec 28 2016

Rexx

See Meyer link.

Formula

For any odd a(m), a(m) = A211456(m) + A211457(m). - Thomas Ordowski, Dec 09 2013

Extensions

Used a comment line to give a more explicit definition. - N. J. A. Sloane, Nov 12 2010

Definition corrected by Max Alekseyev, Nov 12 2010