A335336 - OEIS

%I #41 Apr 22 2024 08:12:02

%S 687979968481,1928376089641,2638625591701,3148470889201,3152088903601,

%T 14682521533681,19656816822721,37333372057201,47003559452641,

%U 80643055074121,129235662445121,140940741166849,196945133626801,336301807660741,345186571310209,439931062854361

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

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

%C 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.

%H Amiram Eldar, <a href="/A335336/b335336.txt">Table of n, a(n) for n = 1..3150</a> (terms below 10^22, calculated using data from Claude Goutier; terms 1..459 from Daniel Suteu)

%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.

%H Richard J. McIntosh and Mitra Dipra, <a href="https://doi.org/10.1016/j.jnt.2014.06.005">Carmichael numbers with p+1|n+1</a>, Journal of Number Theory, Volume 147, February 2015, Pages 81-91.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael_number">Carmichael number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas%E2%80%93Carmichael_number">Lucas-Carmichael number</a>.

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%e 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.

%Y Cf. A002997, A006530, A006972, A329948.

%K nonn

%O 1,1

%A _Daniel Suteu_, Jun 04 2020