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A352134
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Numbers k such that the centered cube number k^3 + (k+1)^3 is equal to at least one other sum of two cubes.
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18
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
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FORMULA
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EXAMPLE
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3 belongs to the sequence as 3^3 + 4^3 = 6^3 + (-5)^3 = 91.
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.
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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)
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PROG
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(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
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352135, A352136, A352220, A352221, A352222, A352223, A352224, A352225.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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