A033502 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.

1729, 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921, 11346205609, 13079177569, 21515221081, 27278026129, 65700513721, 71171308081, 100264053529, 168003672409, 172018713961, 173032371289, 464052305161

Offset: 1

Author

Marc LeBrun

Keywords

• nonn

Comments

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]

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

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

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

References

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

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Vol. 45, No. 4 (1939), pp. 269-274.
• Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
• Douglas E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, Vol. 18 (2018), #77. See p. 9.
• Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
• Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
• G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens, Vol. 6 (1899), pp. 142-144.
• Eric Weisstein's World of Mathematics, Carmichael Number.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

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

Cf. A242980, A242981.

Programs

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

Mathematica

CarmichaelNbrQ[n_] := ! PrimeQ@ n && Mod[n, CarmichaelLambda@ n] == 1; (6# + 1)(12# + 1)(18# + 1) & /@
Select[ Range@ 1000, PrimeQ[6# + 1] && PrimeQ[12# + 1] && PrimeQ[18# + 1] && CarmichaelNbrQ[(6# + 1)(12# + 1)(18# + 1)] &]

Extensions

Definition corrected (thanks to Umberto Cerruti) by Bruno Berselli, Jan 18 2013