OFFSET
1,1
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
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
EXAMPLE
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)
12 123 3767491 160 -69
13 163 8741691 206 -5 (206 < 254)
(End)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Pletser, Mar 05 2022
EXTENSIONS
Missing terms inserted by Jon E. Schoenfield, Mar 09 2022
STATUS
approved