A335336 Carmichael numbers k such that k+1 is divisible by gpf(k)+1, where gpf = A006530.

687979968481, 1928376089641, 2638625591701, 3148470889201, 3152088903601, 14682521533681, 19656816822721, 37333372057201, 47003559452641, 80643055074121, 129235662445121, 140940741166849, 196945133626801, 336301807660741, 345186571310209, 439931062854361

Offset: 1

Author

Daniel Suteu, Jun 04 2020

Keywords

• nonn

Comments

Are there any Carmichael numbers k with exactly four prime factors such that k+1 is divisible by gpf(k)+1?

Richard J. McIntosh and Mitra Dipra found the following base 2 Fermat pseudoprimes with exactly four prime factors satisfying s-1 | k-1 and s+1 | k+1, where s is the largest prime factor of k: 988679226253951, 3143193486942417481, 44307784380481317090001.

Examples

			For k = 687979968481 = 13 * 29 * 71 * 181 * 211 * 673, which is a Carmichael number, we have gpf(k) = 673. Thereafter we have gpf(k)+1 = 2 * 337 and k+1 = 2 * 337 * 347 * 911 * 3229, satisfying gpf(k)+1 | k+1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..3150 (terms below 10^22, calculated using data from Claude Goutier; terms 1..459 from Daniel Suteu)
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Richard J. McIntosh and Mitra Dipra, Carmichael numbers with p+1|n+1, Journal of Number Theory, Volume 147, February 2015, Pages 81-91.
• Wikipedia, Carmichael number.
• Wikipedia, Lucas-Carmichael number.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A002997, A006530, A006972, A329948.