cp's OEIS Frontend

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

Subsequence of A080747 from which this differs for the first time at n=78, with A080747(78) = 294409, a term not present here.

Is this sequence infinite?

According to Sierpinski there are infinitely many Poulet numbers which are not super-Poulet numbers. But his definition of Poulet numbers includes the even pseudoprimes to base 2 (A006935), and the proof is based on the infinitude of this sequence and that super-Poulet numbers are never even.