OFFSET
1,1
COMMENTS
Below 10^22 there are only 2 Carmichael numbers that are the sum of two positive cubes in two or more different ways (i.e., in A001235): 1729 = 1^3 + 12^3 = 9^3 + 10^3 and 23226658794001 = 9001^3 + 28230^3 = 19108^3 + 25329^3.
Chernick's Carmichael numbers (A033502) are 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 (k is a term of A046025). There are no Chernick's Carmichael numbers other than 1729 that are the sum of two positive cubes in two or more different ways (Lagarias, 2018). In the solution to Lagarias's problem it is noted that John P. Robertson showed that if there are Chernick's Carmichael numbers other than 1729 (corresponding to k = 1) that are the sum of two positive cubes (i.e., terms of this sequence), then they have k > 10^5000.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..831 (terms below 10^22)
Jeffrey C. Lagarias, Problem 12048, Problems and Solutions, The American Mathematical Monthly, Vol. 125, No. 6 (2018), p. 562; JSTOR link; Carmichael in a Taxicab, Solution to Problem 12048 by Albert Stadler, ibid., Vol. 127, No. 1 (2020), p. 93; JSTOR link.
Samuel S. Wagstaff, Ramanujan's taxicab number and its ilk, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; ResearchGate link, author's copy.
MATHEMATICA
carmQ[n_] := CompositeQ[n] && Divisible[n-1, CarmichaelLambda[n]]; Select[Range[200000], carmQ[#] && Length[PowersRepresentations[#, 2, 3]] > 0 &]
PROG
(PARI) isA003325(n) = #select(v->min(v[1], v[2])>0, thue(thueinit('z^3+1); , n)) > 0; \\ Charles R Greathouse IV at A003325
is(n) = (n > 1) && !isprime(n) && !((n-1) % lcm(znstar(n)[2])) && isA003325(n);
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 29 2024
STATUS
approved