OFFSET
1,1
COMMENTS
Fermat pseudoprimes to base 2 are also called Poulet numbers.
The most of the terms shown can be written in one of the following two ways:
(1) p*((n + 1)*p - n*p)*((m + 1)*p - m*p);
(2) p*((n*p - (n + 1)*p)*(m*p - (m + 1)*p),
where p is the smallest of the three prime factors and n, m natural numbers.
Exempli gratia for Poulet numbers from 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 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
MATHEMATICA
Select[Range[10^5], PrimeNu[#] == 3 && PowerMod[2, (# - 1), #] == 1 &] (* Amiram Eldar, Jun 28 2019 *)
PROG
(PARI) is(n)=Mod(2, n)^n==2 && bigomega(n)==3 \\ Charles R Greathouse IV, Dec 07 2014
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Marius Coman, Aug 20 2012
STATUS
approved