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

%I A290481 #19 Jan 31 2024 09:52:31
%S A290481 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,
%T A290481 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,
%U A290481 9,20,2,20,19,2,6,8,2,9,8,12,3,1,10,14,10,3
%N A290481 The number of 3-Carmichael numbers that are divisible by the n-th odd prime.
%C A290481 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.
%C A290481 The terms were calculated using Pinch's tables of Carmichael numbers (see link below).
%D A290481 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.
%H A290481 Max Alekseyev, <a href="/A290481/b290481.txt">Table of n, a(n) for n = 1..1000</a>
%H A290481 R. G. E. Pinch, <a href="http://s369624816.websitehome.co.uk/rgep/carpsp.html">Tables relating to Carmichael numbers</a>.
%H A290481 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_019.htm">Conjecture 19, A bound to the largest prime factor of certain Carmichael numbers</a>, The Prime Puzzles and Problems Connection.
%e A290481 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.
%Y A290481 Cf. A065091 (Odd primes), A087788 (3-Carmichael numbers).
%K A290481 nonn
%O A290481 1,2
%A A290481 _Amiram Eldar_, Aug 03 2017