A290481 The number of 3-Carmichael numbers that are divisible by the n-th odd prime.

1, 3, 6, 1, 8, 5, 4, 2, 4, 9, 8, 9, 12, 3, 3, 1, 16, 4, 7, 11, 2, 2, 5, 8, 4, 6, 3, 12, 6, 8, 11, 5, 6, 2, 11, 14, 8, 2, 3, 4, 15, 6, 11, 1, 9, 22, 5, 4, 7, 2, 5, 15, 8, 6, 4, 4, 21, 9, 10, 2, 5, 12, 9, 20, 2, 20, 19, 2, 6, 8, 2, 9, 8, 12, 3, 1, 10, 14, 10, 3

Offset: 1

Author

Amiram Eldar, Aug 03 2017

Keywords

• nonn

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 the number of 3-Carmichael numbers that are divisible by a fixed prime is finite.

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

Examples

			There is only one 3-Carmichael number that is divisible by 3 (561); there are three that are divisible by 5 (1105, 2465 and 10585) and six that are divisible by 7 (1729, 2821, 6601, 8911, 15841 and 52633). Thus a(1)=1, a(2)=3 and a(3)=6.
		

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

• Max Alekseyev, Table of n, a(n) for n = 1..1000
• 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).