OFFSET
1,1
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 Lenstra, H. W. 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)?
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
EXAMPLE
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.
PROG
(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);
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Daniel Suteu, Nov 08 2019
STATUS
approved