A215672 Fermat pseudoprimes to base 2 with three prime factors.

561, 645, 1105, 1729, 1905, 2465, 2821, 4371, 6601, 8481, 8911, 10585, 12801, 13741, 13981, 15841, 16705, 25761, 29341, 30121, 30889, 33153, 34945, 41665, 46657, 52633, 57421, 68101, 74665, 83665, 87249, 88561, 91001, 93961, 113201, 115921, 121465, 137149

Offset: 1

Author

Marius Coman, Aug 20 2012

Keywords

• nonn

Comments

Fermat pseudoprimes to base 2 are also called Poulet numbers.

Most of the terms shown can be written in one of the following two ways:

(1) p*(p*(n + 1) - n)*(p*(m + 1) - m);

(2) p*(p*n - (n + 1))*(p*m - (m + 1)),

where p is the smallest of the three prime factors and n, m natural numbers.

Exempli gratia for Poulet numbers from the first category:

10585 = 5*29*73 = 5*(5*7 - 6)*(5*18 - 17);

13741 = 7*13*151 = 7*(7*2 - 1)*(7*25 - 24);

13981 = 11*31*41 = 11*(11*3 - 2)*(11*4 - 3);

29341 = 13*37*61 = 13*(13*3 - 2)*(13*5 - 4);

137149 = 23*67*89 = 23*(23*3 - 2)*(23*4 - 3).

Exempli gratia for Poulet numbers from the second category:

6601 = 7*23*41 = 7*(7*4 - 5)*(7*7 - 8).

Note: from the numbers from the sequence above, just the numbers 30889, 88561 and 91001 can't be written in one of the two ways.

What these three numbers have in common: they all have a prime divisor q of the form 30*k + 23 (i.e. 23, 53, 83) and can be written as q*((r + 1)*q - r), where r is a natural number.

Conjecture: Any Poulet number P with three or more prime divisors has at least one prime divisor q for that can be written as P = q*((r + 1)*q - r), where r is a natural number.

Note: it can be proved that a Carmichael number can be written this way for any of its prime divisors - see the sequence A213812.

Note: there are also many Poulet numbers with two prime divisors that can be written this way, but here are few exceptions: 7957, 23377, 42799, 49981, 60787.

The conjecture fails for a(80) = 617093 = 43 * 113 * 127. - Charles R Greathouse IV, Dec 07 2014

First differs from A074380 at n=56. - Amiram Eldar, Jun 28 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Poulet Number
• Eric Weisstein's World of Mathematics, Carmichael Number

Crossrefs

The even terms form A296117.

Cf. A001567, A007304, A074380, A080747, A213812.

Programs

Mathematica

Select[Range[10^5], PrimeNu[#] == 3 && PowerMod[2, (# - 1), #] == 1 &] (* Amiram Eldar, Jun 28 2019 *)

PARI

is(n)=Mod(2,n)^n==2 && bigomega(n)==3 \\ Charles R Greathouse IV, Dec 07 2014