A280199 -id:A280199 - cp's OEIS frontend

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

Karsten Meyer, Nov 12 2010

nonn

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

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

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

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

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

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 *)

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

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.
```

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

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

A280196 Numbers n such that a^(n-1) == 1 (mod n^2) has no solutions with 1 < a < n^2-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138

Offset: 1

Robert Israel and Thomas Ordowski, Dec 28 2016

nonn

1 and numbers n such that A185103(n) = n^2 + (-1)^n.
Complement of A280199.
Union of A000244 and A209211.

			a(4) = 4 is in the sequence because a^3 == 1 (mod 4^2) has no solutions except a == 1 (mod 4^2).
a(7) = 9 is in the sequence because a^8 == 1 (mod 9^2) has no solutions except a == 1 (mod 9^2) and a == 80 (mod 9^2), and 80 = 9^2-1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000244, A185103, A209211, A280199.

Aeven:= select(t -> igcd(t-1, numtheory:-phi(t^2))=1, {seq(i,i=2..1000,2}}):
Aodd:= {seq(3^i,i=0..floor(log[3](1000)))}:
sort(convert(Aeven union Aodd, list));

Aeven = Select[Range[2, 1000, 2], GCD[#-1,EulerPhi[#^2]] == 1&];
Aodd = 3^Range[0, Floor[Log[3, 1000]]];
Union[Aeven, Aodd] (* Jean-François Alcover, Apr 27 2019, from Maple *)

cp's OEIS Frontend

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

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