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A085323 Numbers k such that both k and k+1 are sums of two positive cubes.

%I A085323 #29 May 09 2025 07:12:16
%S A085323 854,4940,9603,10744,17919,29743,62558,79001,133273,164304,193192,
%T A085323 205406,214984,242648,263871,378936,431999,447336,488375,517427,
%U A085323 610687,731158,762047,1000511,1061550,1125207,1134124,1157632,1158137,1179520
%N A085323 Numbers k such that both k and k+1 are sums of two positive cubes.
%C A085323 There are 664 terms < 8*10^9, a(664)=7999968373. - _Zak Seidov_, Jul 24 2009
%C A085323 This is an infinite sequence. To see why, consider the (N,N+1) pair N = 16*k^6 - 12*k^4 + 6*k^2 - 2 = (2*k^2 - k - 1)^3 + (2*k^2 + k -1)^3 and N + 1 = 16*k^6 - 12*k^4 + 6*k^2 - 1 = (2*k^2)^3 + (2*k^2 - 1)^3. - _Ant King_, Sep 20 2013
%C A085323 1796609972023999 is the smallest k such that k, k+1, and k+2 are sums of two positive cubes: k = 98978^3 + 93863^3, k+1 = 108620^3 + 80160^3, and k+2 = 120449^3 + 36628^3. - _Giovanni Resta_, May 07 2025
%H A085323 Zak Seidov, <a href="/A085323/b085323.txt">Table of n, a(n) for n = 1..664</a>
%e A085323 854 = 9^3 + 5^3 and 855 = 8^3 + 7^3;
%e A085323 4940 = 17^3 + 3^3 and 4941 = 13^3 + 14^3.
%t A085323 m=110; k=3; t=Union[Flatten[Table[Table[w^k+q^k, {w, q, m}], {q, 1, m}]]]; dt=Delete[ -RotateRight[t]+t, 1]; p=Part[t, Flatten[Position[dt, 1]]]; Select[p, # <= m^3 + 1 &] (* corrected by _Giovanni Resta_, May 09 2025 *)
%Y A085323 Cf. A003325.
%K A085323 nonn
%O A085323 1,1
%A A085323 _Labos Elemer_, Jul 01 2003
%E A085323 Corrected and extended by _Zak Seidov_, Jul 24 2009
%E A085323 Name and Example edited by _Jon E. Schoenfield_, Jul 29 2017