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A352134 Numbers k such that the centered cube number k^3 + (k+1)^3 is equal to at least one other sum of two cubes.

%I A352134 #44 Feb 16 2025 08:34:03
%S A352134 3,4,9,18,32,36,46,58,107,108,121,123,163,197,235,301,393,411,438,481,
%T A352134 490,528,562,607,633,640,804,1090,1111,1128,1293,1374,1436,1517,1524,
%U A352134 1538,1543,1698,2018,2047,2361,3032,3152,3280,3321,4131,4995,5092,5659,5687,5700
%N A352134 Numbers k such that the centered cube number k^3 + (k+1)^3 is equal to at least one other sum of two cubes.
%C A352134 The centered cube number a(n)^3 + (a(n) + 1)^3 is equal to at least one other sum of two cubes: a(n)^3 + (a(n) + 1)^3 = b(n)^3 + c(n)^3 = d(n), with b(n) > a(n) and b(n) > |c(n)|, and where b(n)=A352135(n), c(n)=A352136(n) and d(n)=A352133(n).
%C A352134 A number k is a term iff t = k^3 + (k+1)^3 = (2*k + 1)*(k^2 + k + 1) has one or more divisors s < 2*k such that 12*t/s - 3*s^2 is a square. Each such divisor s is the sum of two integers (other than k and k+1) whose cubes sum to t. - _Jon E. Schoenfield_, Mar 09 2022
%H A352134 Chai Wah Wu, <a href="/A352134/b352134.txt">Table of n, a(n) for n = 1..917</a> (terms 1..275 from Vladimir Pletser)
%H A352134 A. Grinstein, <a href="https://web.archive.org/web/20040320144821/http://zadok.org/mattandloraine/1729.html">Ramanujan and 1729</a>, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
%H A352134 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%F A352134 a(n)^3 + (a(n) + 1)^3 = A352135(n)^3 + A352136(n)^3 = A352133(n).
%e A352134 3 belongs to the sequence as 3^3 + 4^3 = 6^3 + (-5)^3 = 91.
%e A352134 From _Jon E. Schoenfield_, Mar 11 2022: (Start)
%e A352134 The table below lists the first 15 pairs of integers (b,c) such that b > c+1 and b^3 + c^3 is a centered cube number k^3 + (k+1)^3 = d.
%e A352134 Note that there are two pairs (b,c) for k=121 and two for k=163. For these and for all numbers k for which there is more than one pair (b,c), the pair with the smallest value of b is chosen as the one whose values (b,c) appear in A352135 and A352136, i.e., A352135(n) and A352136(n) are the values (b,c) in that pair whose value of b is smallest.
%e A352134 Thus, the 15 solutions listed in the table account for only the first 13 terms of this sequence and of A352133, A352135, and A352136.
%e A352134 .
%e A352134    n  a(n)=k     d(n)  b(n)  c(n)
%e A352134   --  ------  -------  ----  ----
%e A352134    1       3       91     6    -5
%e A352134    2       4      189     6    -3
%e A352134    3       9     1729    12     1
%e A352134    4      18    12691    28   -21
%e A352134    5      32    68705    41    -6
%e A352134    6      36    97309    46    -3
%e A352134    7      46   201159   151  -148
%e A352134    8      58   400491    90   -69
%e A352134    9     107  2484755   171  -136
%e A352134   10     108  2554741   181  -150
%e A352134   11     121  3587409   153    18  (153 < 369)
%e A352134    *     121  3587409   369  -360  ((b,c) omitted from A352135,A352136)
%e A352134   12     123  3767491   160   -69
%e A352134   13     163  8741691   206    -5  (206 < 254)
%e A352134    *     163  8741691   254  -197  ((b,c) omitted from A352135,A352136)
%e A352134 (End)
%o A352134 (Magma) a:=[]; for k in [1..5700] do t:=k^3+(k+1)^3; for s in Divisors(t) do if s gt 2*k then break; end if; if IsSquare(12*(t div s) - 3*s^2) then a[#a+1]:=k; break; end if; end for; end for; a; // _Jon E. Schoenfield_, Mar 09 2022
%Y A352134 Cf. A005898, A001235, A272885, A352133, A352135, A352136, A352220, A352221, A352222, A352223, A352224, A352225.
%K A352134 nonn
%O A352134 1,1
%A A352134 _Vladimir Pletser_, Mar 05 2022
%E A352134 Missing terms inserted by _Jon E. Schoenfield_, Mar 09 2022