A329223 Poulet numbers (Fermat pseudoprimes to base 2) that are congruent to either 3 or 27 (mod 80) and each prime factor is congruent to 3 mod 80.

51962615262396907, 330468624532072027, 2255490055253468347, 18436227497407654507

Offset: 1

Author

Daniel Suteu, Nov 08 2019

Keywords

• nonn
• more

Comments

If a term of this sequence is also a Carmichael number (A002997) and a Lucas-Carmichael number (A006972), then it would be a counterexample to Agrawal's conjecture, as Hendrick Lenstra and Carl Pomerance showed.

330468624532072027 is the only Carmichael number below 2^64 that is a term of this sequence. However, it is not a Lucas-Carmichael number.

The sequence also includes: 68435117188079800987, 164853581396047908970027, 522925572082528736632187, 1820034970687975620484907, 4263739170243679206753787, 4360728281510798266333387, 28541906071781213329174507, 33833150661360980271172507, 84444874320158644422192427, 175352076630428496579381067, 270136290676063386556053067, 615437738523352001584590187, 3408560627000081376639770587, 11260257876970792445537580187.

No term with 5 prime factors (which would be congruent to 3 mod 80) is known to the author.

Are all terms also strong pseudoprimes to base 2 (A001262)?

Examples

			51962615262396907 is a term because it is a Fermat pseudoprime to base 2 and it is congruent to 27 (mod 80) and all of its prime factors (643, 154723, 522306163) are congruent to 3 mod 80.
		

Links

• H. W. Lenstra, and Carl Pomerance, Remarks on Agrawal's conjecture, American Institute of Mathematics (2003), pp. 30-32.
• Tomáš Váňa, Agrawal's Conjecture and Carmichael Numbers, student scientific conference, pp. 13-22.
• Wikipedia, Agrawal's conjecture

Crossrefs

Cf. A001567.

Programs

PARI

isok(n) = ((n%80==3) || (n%80==27)) && (Mod(2, n)^(n-1) == 1) || return(0); my(f=factor(n)[,1]); (#f > 1) && (#select(p->p%80==3, f) == #f);