A379656 Carmichael numbers that are the sum of 2 positive cubes.

1729, 15841, 46657, 126217, 188461, 1082809, 1773289, 2628073, 3146221, 5049001, 6868261, 14469841, 19683001, 31146661, 40917241, 78091201, 92625121, 144218341, 252141121, 1836304561, 2616662881, 3035837161, 4354716961, 4828075561, 10779325921, 13200275881, 14235803713

Offset: 1

Author

Amiram Eldar, Dec 29 2024

Keywords

• nonn

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.

Crossrefs

Intersection of A002997 and A003325.

A265628 is a subsequence.

Cf. A001235, A033502, A046025.

Programs

Mathematica

carmQ[n_] := CompositeQ[n] && Divisible[n-1, CarmichaelLambda[n]]; Select[Range[200000], carmQ[#] && Length[PowersRepresentations[#, 2, 3]] > 0 &]

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);