A352134 -id:A352134 - cp's OEIS frontend

A352133 Centered cube numbers that can be written as sums of two other cubes in at least one way.

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

Offset: 1

Vladimir Pletser, Mar 05 2022

nonn

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).

Subsequence of A005898.

			91 belongs to the sequence because 91 = 3^3 + 4^3 = 6^3 + (-5)^3.

Vladimir Pletser, Table of n, a(n) for n = 1..275
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

Cf. A005898, A001235, A272885, A352134, A352135, A352136, A352220, A352221, A352222, A352223, A352224, A352225.

a(n) = A352134(n)^3 + (A352134(n) + 1)^3 = A352135(n)^3 + A352136(n)^3.

A352135 Numbers j in pairs (j,k), with j <> k +- 1, such that the sum of their cubes is equal to a centered cube number.

Original entry on oeis.org

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

Offset: 1

Vladimir Pletser, Mar 05 2022

nonn

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).

			6 belongs to the sequence as 6^3 + (-5)^3 = 3^3 + 4^3 = 91.

Vladimir Pletser, Table of n, a(n) for n = 1..258
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

Cf. A005898, A001235, A272885, A352133, A352134, A352136, A352220, A352221, A352222, A352223, A352224, A352225.

a(n)^3 + A352136(n)^3 = A352134(n)^3 + (A352134(n) + 1)^3 = A352133(n).

Missing terms inserted by Jon E. Schoenfield, Mar 11 2022

A352136 Numbers k in pairs (j,k), with j <> k +- 1, such that the sum of their cubes is equal to a centered cube number.

Original entry on oeis.org

-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

Offset: 1

Vladimir Pletser, Mar 05 2022

sign

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).

			-5 belongs to the sequence as 6^3 + (-5)^3 = 3^3 + 4^3 = 91.

Vladimir Pletser, Table of n, a(n) for n = 1..258
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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352220, A352221, A352222, A352223, A352224, A352225.

A352135(n)^3 + a(n)^3 = A352134(n)^3 + (A352134(n) + 1)^3 = A352133(n).

A352223 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.

Original entry on oeis.org

18, -5, 107, -125, 712, -1152, -1719, -865, -5370, -7870, 2518, -963, -29949, -20030, 111491, 87797, 261536, 2274319, -140357, -3938794, -139674130, -792131385

Offset: 1

Vladimir Pletser, Mar 07 2022

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).
Subsequence of A352136.

			18 belongs to the sequence as 153^3 + 18^3 = 121^3 + 122^3 = 369^3 + (-360)^3 = 3587409.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352224, A352225.

A352223(n)^3 + a(n)^3 = A352221(n)^3 + (A352221(n) + 1)^3 = A352224(n)^3 + A352225(n)^3 = A352220(n).

a(21) from Chai Wah Wu, Mar 17 2022

a(22) from Bert Dobbelaere, Apr 18 2022

A352224 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.

Original entry on oeis.org

369, 254, 419, 2820, 3923, 10090, 29538, 8310, 227835, 20739, 28391, 37494, 875196, 112295, 623814, 478788, 3045867, 17595980, 5473454, 10365237, 13724103165, 94822722216

Offset: 1

Vladimir Pletser, Mar 07 2022

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).

			369 belongs to the sequence as 369^3 + (-360)^3 = 121^3 + 122^3 = 153^3 + 18^3 = 3587409.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352223, A352225.

a(n)^3 + A352225(n)^3 = A352221(n)^3 + (A352221(n) + 1)^3 = A352222(n)^3 + A352223(n)^3 = A352220(n).

a(21) from Chai Wah Wu, Mar 17 2022

a(22) from Bert Dobbelaere, Apr 18 2022

A352225 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.

Original entry on oeis.org

-360, -197, -362, -2805, -3866, -10081, -29511, -5905, -227790, -10012, -24548, -28995, -875133, -73040, -615709, -457027, -3044074, -17549681, -4232837, -4999714, -13724102460, -94822721073

Offset: 1

Vladimir Pletser, Mar 07 2022

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).

			-360 belongs to the sequence as 369^3 + (-360)^3 = 121^3 + 122^3 = 153^3 + 18^3 = 3587409.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352222, A352223, A352224.

A352224(n)^3 + a(n)^3 = A352221(n)^3 + (A352221(n) + 1)^3 = A352222(n)^3 + A352223(n)^3 = A352220(n).

a(21) from Chai Wah Wu, Mar 17 2022

a(22) from Bert Dobbelaere, Apr 18 2022

A352220 Centered cube numbers that can be written as sums of two other cubes in at least two ways.

Original entry on oeis.org

3587409, 8741691, 26122131, 355957875, 2593625571, 2746367559, 70607389041, 367954598375, 7006302268875, 7916366521691, 8091803325879, 28332679374909, 144757538551899, 1026401875608375, 9339629571431315, 14295468330521189, 49873257556492139, 42892025638971003759

Offset: 1

Vladimir Pletser, Mar 07 2022

nonn

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).

Subsequence of A005898 and of A352133.

			3587409 belongs to the sequence because 3587409 = 121^3 + 122^3 = 153^3 + 18^3 = 369^3 + (-360)^3.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225.

a(n) = A352221(n)^3 + (A352221(n) + 1)^3 = A352222(n)^3 + A352223(n)^3 = A352224(n)^3 + A352225(n)^3.

a(6)-a(18) from Jon E. Schoenfield, Mar 09 2022

A352221 Numbers k such that the centered cube number k^3 + (k+1)^3 is equal to at least two other sums of two cubes.

Original entry on oeis.org

121, 163, 235, 562, 1090, 1111, 3280, 5687, 15187, 15818, 15934, 24196, 41674, 80062, 167147, 192629, 292154, 2778319, 3532195, 7906844, 58400437, 248878534

Offset: 1

Vladimir Pletser, Mar 07 2022

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).

Subsequence of A352134.

			121 is a term because 121^3 + 122^3 = 153^3 + 18^3 = 369^3 + (-360)^3 = 3587409.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352222, A352223, A352224, A352225.

a(n)^3 + (a(n)+1)^3 = A352222(n)^3 + A352223(n)^3 = A352224(n)^3 + A352225(n)^3 = A352220(n).

a(6)-a(20) from Jon E. Schoenfield, Mar 10 2022
a(21) from Chai Wah Wu, Mar 17 2022
a(22) from Bert Dobbelaere, Apr 18 2022

A352222 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.

Original entry on oeis.org

153, 206, 292, 710, 1307, 1623, 4230, 7170, 19275, 20331, 20063, 30486, 55572, 101135, 199614, 238806, 317427, 3145700, 4450334, 10163157, 146173525, 808182534

Offset: 1

Vladimir Pletser, Mar 07 2022

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).
Subsequence of A352135.

			153 belongs to the sequence as 153^3 + 18^3 = 121^3 + 122^3 = 369^3 + (-360)^3 = 3587409.

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

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225.

a(n)^3 + A352223(n)^3 = A352221(n)^3 + (A352221(n) + 1)^3 = A352224(n)^3 + A352225(n)^3 = A352220(n).

a(21) from Chai Wah Wu, Mar 17 2022

a(22) from Bert Dobbelaere, Apr 18 2022

A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t(3t^2 + 4)(t^2(3t^2 + 4)^2 + 3)/4 with t = 2n-1 and n > 0.

Original entry on oeis.org

91, 201159, 15407765, 295233841, 2746367559, 16448122691, 73287987409, 264133278045, 811598515091, 2202365761759, 5410166901741, 12249942682409, 25914353312575, 51755729480091, 98389720844009, 179211321358741, 314429627203659, 533744613620855, 879807401606341, 1412624924155809

Offset: 1

Vladimir Pletser, Apr 02 2022

Numbers A > 0 such that A = B^3 + (B+1)^3 = C^3 - D^3 and such that C - D = 2n - 1, with C > D > B > 0, and A = t*(3*t^2 + 4)*(t^2*(3*t^2 + 4)^2 + 3)/4 with t = 2*n-1, and where A = a(n) (this sequence), B = A352756(n), C = A352757(n) and D = A352758(n).
There are infinitely many such numbers a(n) = A in this sequence.
Subsequence of A005898, of A352133 and of A352220.

			a(1) = 91 belongs to the sequence because 91 = 3^3 + 4^3 = 6^3 - 5^3 and 6 - 5 = 1 = 2*1 - 1.
a(2) = 201159 belongs to the sequence because 201159 = 46^3 + 47^3 = 151^3 - 148^3 and 151 - 148 = 3 = 2*2 - 1.
a(3) = (2*3 - 1)*(3*(2*3 - 1)^2 + 4)*((2*3 - 1)^2*(3*(2*3 - 1)^2 + 4)^2 + 3)/4 = 15407765.

Vladimir Pletser, Table of n, a(n) for n = 1..10000
A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Vladimir Pletser, Euler's and the Taxi-Cab relations and other numbers that can be written twice as sums of two cubed integers, submitted. Preprint available on ResearchGate, 2022.
Eric Weisstein's World of Mathematics, Centered Cube Number
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352221, A352222, A352223, A352224, A352225, A352756, A352757, A352758, A352759, A355751, A355752, A355753.

restart; for n to 20 do (1/4)*(2*n-1)*(3*(2*n-1)^2+4)*((2*n-1)^2*(3*(2*n-1)^2+4)^2+3) end do;

a(n) = A352756(n)^3 + (A352756(n) + 1)^3 = A352757(n)^3 - A352758(n)^3 and A352757(n) - A352758(n) = 2n - 1.
a(n) = (2*n - 1)*(3*(2*n - 1)^2 + 4)*((2*n - 1)^2*(3*(2*n - 1)^2 + 4)^2 + 3)/4.
a(n) can be extended for negative n such that a(-n) = -a(n+1).

cp's OEIS Frontend

A352133 Centered cube numbers that can be written as sums of two other cubes in at least one way.

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A352135 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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A352136 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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A352223 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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A352224 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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A352225 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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A352220 Centered cube numbers that can be written as sums of two other cubes in at least two ways.

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A352221 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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A352222 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.

A352755 Positive centered cube numbers that can be written as the difference of two positive cubes: a(n) = t(3t^2 + 4)(t^2(3t^2 + 4)^2 + 3)/4 with t = 2n-1 and n > 0.