%I #53 Aug 04 2022 05:08:11

%S 1729,294409,56052361,118901521,172947529,216821881,228842209,

%T 1299963601,2301745249,9624742921,11346205609,13079177569,21515221081,

%U 27278026129,65700513721,71171308081,100264053529,168003672409,172018713961,173032371289,464052305161

%N 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.

%C Also called Chernick's Carmichael numbers. The polynomial (6*k+1)*(12*k+1)*(18*k+1) is the simplest Chernick polynomial. [Named after the American physicist and mathematician Jack Chernick (1911-1971). - _Amiram Eldar_, Jun 15 2021]

%C The first term, 1729, is the Hardy-Ramanujan number and the smallest primary Carmichael number (A324316).

%C Dickson's conjecture implies that this sequence is infinite, as pointed out by Chernick.

%C All terms of this sequence are primary Carmichael numbers (A324316) having the following remarkable property. Let m be a term of A033502. For each prime divisor p of m, the sum of the base-p digits of m equals p. This property also holds for "almost all" 3-term Carmichael numbers (A087788), since they can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions, see Kellner 2019. - _Bernd C. Kellner_, Aug 03 2022

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A13, pp. 50-53.

%H Donovan Johnson, <a href="/A033502/b033502.txt">Table of n, a(n) for n = 1..10000</a>

%H Jack Chernick, <a href="http://projecteuclid.org/euclid.bams/1183501763">On Fermat's simple theorem</a>, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.

%H Douglas E. Iannucci, <a href="http://math.colgate.edu/~integers/s77/s77.Abstract.html">When the small divisors of a natural number are in arithmetic progression</a>, INTEGERS, Electronic Journal of Combinatorial Number Theory, Vol. 18 (2018), #77. See p. 9.

%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.

%H Bernd C. Kellner, <a href="http://math.colgate.edu/~integers/w38/w38.pdf">On primary Carmichael numbers</a>, Integers 22 (2022), #A38, 39 pp.; arXiv:<a href="https://arxiv.org/abs/1902.11283">1902.11283</a> [math.NT], 2019.

%H G. Tarry, I. Franel, A. Korselt, and G. Vacca. <a href="https://oeis.org/wiki/File:Probl%C3%A8me_chinois.pdf">Problème chinois</a>. L'intermédiaire des mathématiciens, Vol. 6 (1899), pp. 142-144.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%t CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; (6# + 1)(12# + 1)(18# + 1) & /@

%t Select[ Range@ 1000, PrimeQ[6# + 1] && PrimeQ[12# + 1] && PrimeQ[18# + 1] && CarmichaelNbrQ[(6# + 1)(12# + 1)(18# + 1)] &]

%o (Magma) [n : k in [1..710] | IsPrime(a) and IsPrime(b) and IsPrime(c) and IsOne(n mod CarmichaelLambda(n)) where n is a*b*c where a is 6*k+1 where b is 12*k+1 where c is 18*k+1]; // _Arkadiusz Wesolowski_, Oct 29 2013

%Y Values of k are given by A046025. Subsequence of A002997, A087788, and A324316.

%Y Cf. A242980, A242981.

%K nonn

%O 1,1

%A _Marc LeBrun_

%E Definition corrected (thanks to Umberto Cerruti) by _Bruno Berselli_, Jan 18 2013