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

1,1

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

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

Table of n, a(n) for n=1..35.

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.

Example

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

Crossrefs

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

Sequence in context: A267607 A319248 A340865 * A156560 A028342 A137982

Adjacent sequences: A290481 A290482 A290483 * A290485 A290486 A290487

Keyword

nonn,more

Author

Amiram Eldar, Aug 03 2017

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

Status

approved