A290484 Odd prime numbers that are factors of only one 3-Carmichael number.

3, 11, 59, 197, 389, 467, 479, 503, 563, 719, 839, 887, 1523, 1907, 2087, 2339, 2837, 3167, 3989, 4229, 4259, 4643, 4679, 4787, 4903, 4919, 5417, 5849, 5879, 6299, 7307, 7331, 7577, 7583, 8117

Offset: 1

Author

Amiram Eldar, Aug 03 2017

Keywords

• nonn
• more

Comments

Beeger proved in 1950 that if p < q < r are primes such that p*q*r is a 3-Carmichael number, then q < 2p^2 and r < p^3. Therefore there is a finite number of 3-Carmichael numbers which divisible by a given prime.

An odd prime p is a term if and only if A290481(A033270(p)) = 1. - Max Alekseyev, Jan 31 2024

Examples

			59 is in the sequence since it is a prime factor of only one 3-Carmichael number: 178837201 = 59 * 1451 * 2089.
		

References

• N. G. W. H. Beeger, "On composite numbers n for which a^n == 1 (mod n) for every a prime to n", Scripta Mathematica, Vol. 16 (1950), pp. 133-135.

Links

• R. G. E. Pinch, Tables relating to Carmichael numbers.
• Carlos Rivera, Conjecture 19, A bound to the largest prime factor of certain Carmichael numbers, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A065091 (Odd primes), A087788 (3-Carmichael numbers), A051663, A290481, A369777.

Extensions

a(1)-a(12) were calculated using Pinch's tables of Carmichael numbers (see links).

a(13)-a(35) from Max Alekseyev, Jan 31 2024