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All members 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
the following remarkable property. Let m be an element 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
(MAGMAMagma) [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
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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).
All members of this sequence are primary Carmichael numbers (A324316) having
the following remarkable property. Let m be an element 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
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.
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.
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R. Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A13, pp. 50-53.
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.
D. 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, #77, Vol. 18 (2018), #77. See p. 9.
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.
Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.
<a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers.</a>.
Also called Chernick's Carmichael numbers. [Named after the American physicist and mathematician Jack Chernick (1911-1971). - _Amiram Eldar_, Jun 15 2021]
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