Search: a272885 -id:a272885
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A352133
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Centered cube numbers that can be written as sums of two other cubes in at least one way.
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+10
18
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91, 189, 1729, 12691, 68705, 97309, 201159, 400491, 2484755, 2554741, 3587409, 3767491, 8741691, 15407765, 26122131, 54814509, 121861441, 139361059, 168632191, 223264809, 236019771, 295233841, 355957875, 448404255, 508476241, 525518721, 1041378589, 2593625571, 2746367559, 2874318841, 4328420941, 5193550999
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OFFSET
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1,1
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COMMENTS
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Numbers that are the sum of two consecutive cubes and at least one other sum of two cubes: a(n) = b(n)^3 + (b(n) + 1)^3 = c(n)^3 + d(n)^3, with c(n) > b(n) and c(n) > |d(n)|, and where b(n)=A352134(n), c(n)=A352135(n) and d(n)=A352136(n).
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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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91 belongs to the sequence because 91 = 3^3 + 4^3 = 6^3 + (-5)^3.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352134, 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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STATUS
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approved
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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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+10
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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A352135
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Numbers j in pairs (j,k), with j <> k +- 1, such that the sum of their cubes is equal to a centered cube number.
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+10
18
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6, 6, 12, 28, 41, 46, 151, 90, 171, 181, 153, 160, 206, 1016, 292, 378, 513, 531, 831, 633, 618, 3753, 710, 1119, 1410, 830, 1246, 1307, 1623, 1506, 1629, 1752, 1845, 1917, 1917, 2019, 10815, 2140, 22331, 2871, 3660, 4481, 3881, 4230, 43356, 9955, 6294, 76621, 22988, 7170, 21253
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OFFSET
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1,1
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COMMENTS
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Numbers j such that j^3 + k^3 = m^3 + (m + 1)^3 = N, with j <> (k +- 1), j > m and j > |k|, and where j = a(n) (this sequence), k = A352136(n), m = A352134(n) and N = A352133(n).
In case there are two or more pairs of numbers (j, k) such that the sum of their cubes equals the same centered cube number, the smallest occurrence of j is shown in the sequence. For other occurrences, see A352224(n) and A352225(n).
Terms in Data are ordered according to increasing order of A352133(n) or A352134(n).
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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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6 belongs to the sequence as 6^3 + (-5)^3 = 3^3 + 4^3 = 91.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, 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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A352136
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Numbers k in pairs (j,k), with j <> k +- 1, such that the sum of their cubes is equal to a centered cube number.
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+10
18
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-5, -3, 1, -21, -6, -3, -148, -69, -136, -150, 18, -69, -5, -1011, 107, 93, -236, -218, -740, -312, -21, -3746, -125, -984, -1319, -359, -963, 712, -1152, -815, 178, -569, -706, -382, 346, -982, -10794, -69, -22320, -1866, -2831, -3246, 1614, -1719, -43343, -9456, -197, -76606, -22757, -865, -20976
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OFFSET
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1,1
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COMMENTS
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Numbers k such that j^3 +k^3 = m^3 + (m + 1)^3 = N, with j <> (k +- 1), j > m and j > |k|, and where j = A352135(n), k = a(n) (this sequence), m = A352134(n) and N = A352133(n).
In case there are two or more pairs of numbers (j, k) such that the sum of their cubes equals the same centered cube number, the smallest occurrence of j is shown in the sequence. For other occurrences, see A352224(n) and A352225(n).
Terms in Data are ordered according to increasing order of A352133(n) or A352134(n).
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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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-5 belongs to the sequence as 6^3 + (-5)^3 = 3^3 + 4^3 = 91.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352220, A352221, A352222, A352223, A352224, A352225.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A352223
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Second members D of two non-consecutive numbers such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e., A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.
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+10
18
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18, -5, 107, -125, 712, -1152, -1719, -865, -5370, -7870, 2518, -963, -29949, -20030, 111491, 87797, 261536, 2274319, -140357, -3938794, -139674130, -792131385
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OFFSET
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1,1
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COMMENTS
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Numbers D such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = A352221(n), C = A352222(n), D = a(n) (this sequence), E = A352224(n) and F = A352225(n).
Terms in Data are ordered according to increasing order of A352220(n) or A352221(n).
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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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18 belongs to the sequence as 153^3 + 18^3 = 121^3 + 122^3 = 369^3 + (-360)^3 = 3587409.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225.
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KEYWORD
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sign,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A352224
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First numbers E = a(n) of two non-consecutive numbers (E, F) different from (C, D) = (A352222(n), A352223(n)), such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e. A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.
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+10
17
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369, 254, 419, 2820, 3923, 10090, 29538, 8310, 227835, 20739, 28391, 37494, 875196, 112295, 623814, 478788, 3045867, 17595980, 5473454, 10365237, 13724103165, 94822722216
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OFFSET
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1,1
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COMMENTS
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Numbers E such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = A352221(n), C = A352222(n), D = A352223(n), E = a(n) (this sequence) and F = A352225(n).
Terms are ordered according to increasing order of A352220(n) or A352221(n).
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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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369 belongs to the sequence as 369^3 + (-360)^3 = 121^3 + 122^3 = 153^3 + 18^3 = 3587409.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352223, A352225.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A352225
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Second numbers F = a(n) of two non-consecutive numbers (E, F) different from (C, D) = (A352222(n), A352223(n)), such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e. A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.
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+10
17
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-360, -197, -362, -2805, -3866, -10081, -29511, -5905, -227790, -10012, -24548, -28995, -875133, -73040, -615709, -457027, -3044074, -17549681, -4232837, -4999714, -13724102460, -94822721073
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OFFSET
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1,1
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COMMENTS
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Numbers F such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = A352221(n), C = A352222(n), D = A352223(n), E = A352224(n) and F = a(n) (this sequence).
Terms are ordered according to increasing order of A352220(n) or A352221(n).
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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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-360 belongs to the sequence as 369^3 + (-360)^3 = 121^3 + 122^3 = 153^3 + 18^3 = 3587409.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352223, A352224.
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KEYWORD
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sign,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A352220
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Centered cube numbers that can be written as sums of two other cubes in at least two ways.
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+10
16
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3587409, 8741691, 26122131, 355957875, 2593625571, 2746367559, 70607389041, 367954598375, 7006302268875, 7916366521691, 8091803325879, 28332679374909, 144757538551899, 1026401875608375, 9339629571431315, 14295468330521189, 49873257556492139, 42892025638971003759
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OFFSET
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1,1
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COMMENTS
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Numbers A such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = a(n) (this sequence), B = A352221(n), C = A352222(n), D = A352223(n), E = A352224(n) and F = A352225(n).
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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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3587409 belongs to the sequence because 3587409 = 121^3 + 122^3 = 153^3 + 18^3 = 369^3 + (-360)^3.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, 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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A352221
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Numbers k such that the centered cube number k^3 + (k+1)^3 is equal to at least two other sums of two cubes.
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+10
16
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121, 163, 235, 562, 1090, 1111, 3280, 5687, 15187, 15818, 15934, 24196, 41674, 80062, 167147, 192629, 292154, 2778319, 3532195, 7906844, 58400437, 248878534
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OFFSET
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1,1
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COMMENTS
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Numbers B such that the centered cube number B^3 + (B+1)^3 is equal to at least two other sums of two cubes, i.e., A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = a(n) (this sequence), C = A352222(n), D = A352223(n), E = A352224(n) and F = A352225(n).
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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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121 is a term because 121^3 + 122^3 = 153^3 + 18^3 = 369^3 + (-360)^3 = 3587409.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A352222
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First members C of two non-consecutive numbers such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e., A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.
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+10
15
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153, 206, 292, 710, 1307, 1623, 4230, 7170, 19275, 20331, 20063, 30486, 55572, 101135, 199614, 238806, 317427, 3145700, 4450334, 10163157, 146173525, 808182534
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OFFSET
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1,1
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COMMENTS
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Numbers C such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = A352221(n), C = a(n) (this sequence), D = A352223(n), E = A352224(n) and F = A352225(n).
Terms are ordered according to increasing order of A352220(n) or A352221(n).
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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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153 belongs to the sequence as 153^3 + 18^3 = 121^3 + 122^3 = 369^3 + (-360)^3 = 3587409.
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CROSSREFS
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Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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