

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.


17



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

1,1


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 (19111971).  Amiram Eldar, Jun 15 2021]
The first term, 1729, is the HardyRamanujan 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 basep digits of m equals p. This property also holds for "almost all" 3term Carmichael numbers (A087788), since they can be represented by certain Chernick polynomials, whose values obey a strict sdecomposition (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. 5053.


LINKS

G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens, Vol. 6 (1899), pp. 142144.


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)] &]


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



