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.

3, 4, 9, 18, 32, 36, 46, 58, 107, 108, 121, 123, 163, 197, 235, 301, 393, 411, 438, 481, 490, 528, 562, 607, 633, 640, 804, 1090, 1111, 1128, 1293, 1374, 1436, 1517, 1524, 1538, 1543, 1698, 2018, 2047, 2361, 3032, 3152, 3280, 3321, 4131, 4995, 5092, 5659, 5687, 5700

Offset: 1

Author

Vladimir Pletser, Mar 05 2022

Keywords

• nonn

Comments

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).

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

Examples

			3 belongs to the sequence as 3^3 + 4^3 = 6^3 + (-5)^3 = 91.
From _Jon E. Schoenfield_, Mar 11 2022: (Start)
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.
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.
Thus, the 15 solutions listed in the table account for only the first 13 terms of this sequence and of A352133, A352135, and A352136.
.
   n  a(n)=k     d(n)  b(n)  c(n)
  --  ------  -------  ----  ----
   1       3       91     6    -5
   2       4      189     6    -3
   3       9     1729    12     1
   4      18    12691    28   -21
   5      32    68705    41    -6
   6      36    97309    46    -3
   7      46   201159   151  -148
   8      58   400491    90   -69
   9     107  2484755   171  -136
  10     108  2554741   181  -150
  11     121  3587409   153    18  (153 < 369)
   *     121  3587409   369  -360  ((b,c) omitted from A352135,A352136)
  12     123  3767491   160   -69
  13     163  8741691   206    -5  (206 < 254)
   *     163  8741691   254  -197  ((b,c) omitted from A352135,A352136)
(End)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..917 (terms 1..275 from Vladimir Pletser)
• A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
• Eric Weisstein's World of Mathematics, Centered Cube Number

Crossrefs

Cf. A005898, A001235, A272885, A352133, A352135, A352136, A352220, A352221, A352222, A352223, A352224, A352225.

Programs

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

Formula

a(n)^3 + (a(n) + 1)^3 = A352135(n)^3 + A352136(n)^3 = A352133(n).

Extensions

Missing terms inserted by Jon E. Schoenfield, Mar 09 2022