This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379656 #16 Dec 29 2024 08:50:12
%S A379656 1729,15841,46657,126217,188461,1082809,1773289,2628073,3146221,
%T A379656 5049001,6868261,14469841,19683001,31146661,40917241,78091201,
%U A379656 92625121,144218341,252141121,1836304561,2616662881,3035837161,4354716961,4828075561,10779325921,13200275881,14235803713
%N A379656 Carmichael numbers that are the sum of 2 positive cubes.
%C A379656 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.
%C A379656 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.
%H A379656 Amiram Eldar, <a href="/A379656/b379656.txt">Table of n, a(n) for n = 1..831</a> (terms below 10^22)
%H A379656 Jeffrey C. Lagarias, <a href="https://doi.org/10.1080/00029890.2018.1460990">Problem 12048</a>, Problems and Solutions, The American Mathematical Monthly, Vol. 125, No. 6 (2018), p. 562; <a href="https://www.jstor.org/stable/48662615">JSTOR link</a>; <a href="https://doi.org/10.1080/00029890.2020.1678347">Carmichael in a Taxicab</a>, Solution to Problem 12048 by Albert Stadler, ibid., Vol. 127, No. 1 (2020), p. 93; <a href="https://www.jstor.org/stable/48661468">JSTOR link</a>.
%H A379656 Samuel S. Wagstaff, <a href="https://doi.org/10.1007/s11139-024-00846-1">Ramanujan's taxicab number and its ilk</a>, The Ramanujan Journal, Vol. 64, No. 3 (2024), pp. 761-764; <a href="https://www.researchgate.net/publication/379960100_Ramanujan's_taxicab_number_and_its_ilk">ResearchGate link</a>, <a href="https://homes.cerias.purdue.edu/~ssw/taxi.pdf">author's copy</a>.
%t A379656 carmQ[n_] := CompositeQ[n] && Divisible[n-1, CarmichaelLambda[n]]; Select[Range[200000], carmQ[#] && Length[PowersRepresentations[#, 2, 3]] > 0 &]
%o A379656 (PARI) isA003325(n) = #select(v->min(v[1], v[2])>0, thue(thueinit('z^3+1);, n)) > 0; \\ _Charles R Greathouse IV_ at A003325
%o A379656 is(n) = (n > 1) && !isprime(n) && !((n-1) % lcm(znstar(n)[2])) && isA003325(n);
%Y A379656 Intersection of A002997 and A003325.
%Y A379656 A265628 is a subsequence.
%Y A379656 Cf. A001235, A033502, A046025.
%K A379656 nonn
%O A379656 1,1
%A A379656 _Amiram Eldar_, Dec 29 2024