cp's OEIS Frontend

Carmichael numbers are composite numbers n such that k = 1 (mod lambda(k)); equivalently, lambda(k)^2 divides (k-1)^2. As a result, all composite terms of the sequence are Carmichael numbers A002997. But there are no primes in this sequence except for 2 (since lambda(p) = p-1 and (p-1)^3 > (p-1)^2 for p > 2) and so all terms in this sequence other than 1 and 2 are Carmichael numbers. - Charles R Greathouse IV, Oct 15 2016

Is this sequence infinite?

Corresponding Fermat pseudoprimes to base 2 are 1729, 46657, 2628073, 19683001, 162771337, 110592000001, 432081216001, ...

There is only one composite term up to 10^10: 14701. It also appears in A265628 (see comments). Can we say that if there is a Fermat pseudoprime to base 2 of the form (k-1)^3 + 1, k is a prime number most of the time? Are there other composite terms like 14701?

Also, Carmichael numbers of the form k^2 + k + 1.

Also, Carmichael numbers of the form k^2 - k + 1.

There are no other terms below 10^22.

Carmichael numbers m such that 4m - 3 is square. - Thomas Ordowski, Apr 30 2018

Below 10^22 there are only 2 Carmichael numbers that are the sum of two positive cubes in two or more different ways (i.e., in A001235): 1729 = 1^3 + 12^3 = 9^3 + 10^3 and 23226658794001 = 9001^3 + 28230^3 = 19108^3 + 25329^3.

Chernick's Carmichael numbers (A033502) are Carmichael numbers of the form (6*k+1)*(12*k+1)*(18*k+1), where 6*k+1, 12*k+1 and 18*k+1 are all primes (k is a term of A046025). There are no Chernick's Carmichael numbers other than 1729 that are the sum of two positive cubes in two or more different ways (Lagarias, 2018). In the solution to Lagarias's problem it is noted that John P. Robertson showed that if there are Chernick's Carmichael numbers other than 1729 (corresponding to k = 1) that are the sum of two positive cubes (i.e., terms of this sequence), then they have k > 10^5000.