A343081 -id:A343081 - cp's OEIS frontend

A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

1729, 251, 219, 157, 158, 131, 132, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126

Offset: 2

N. J. A. Sloane, Apr 03 2021

nonn

This is r(n,3,2) in Alter's notation.

			a(2) = 1729 = 12^3 + 1^3 = 10^3 + 9^3 (the famous Hardy-Ramanujan number).
a(3) = 251 = 5^3 + 5^3 + 1^3 = 6^3 + 3^3 + 2^3.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 2, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001235, A011541, A230563, A342903, A343077, A343078, A343079, A343081, A343085.

a(n) = n+63 for n >= 9.

A343082 a(n) is the smallest number that is the sum of n positive 4th powers in three ways.

Original entry on oeis.org

811538, 16578, 4225, 2676, 2677, 518, 519, 520, 521, 522, 523, 524, 525, 526, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,4,3) in Alter's notation.

			a(3) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.
a(4) = 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 8, page 192.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A085559, A123075, A342893, A343077, A343081, A343083, A343086.

a(n) = n + 255 for n >= 17.

A343083 a(n) is the smallest number that is the sum of n positive 5th powers in three ways.

Original entry on oeis.org

13124675, 696467, 84457, 52417, 52418, 8194, 8195, 8196, 8197, 8198, 8199, 8200, 8201, 8202, 8203, 8204, 7796, 7797, 7798, 7799, 7800, 7801, 7802, 7585, 7586, 7587, 7533, 7534, 7535, 7536, 7537, 4128, 4129, 4130, 4131, 4132, 4133, 4134, 4135, 4136, 4137, 4138, 4139, 4140, 4141, 4142, 4143, 4144, 4145, 4146, 4147, 4148, 4149, 4150, 4151, 4152, 4153, 4154, 4155, 4156, 4157, 2112

Offset: 5

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,5,3) in Alter's notation.

			a(5) = 13124675 = 1^5 + 9^5 + 10^5 + 20^5 + 25^5 = 2^5 + 5^5 + 12^5 +23^5 + 23^5 = 16^5 + 19^5 + 20^5 + 20^5 + 20^5.
a(6) = 696467 = 1^5 + 6^5 + 8^5 + 9^5 + 9^5 + 14^5 = 3^5 + 3^5 + 7^5 + 9^5 + 12^5 + 13^5 = 4^5 + 4^5 + 4^5 + 11^5 + 11^5 + 13^5.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 9, page 193.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342895, A343077, A343080, A343081, A343082.

a(n) = n + 2046 for n >= 66.

A343085 a(n) is the smallest number that is the sum of n positive cubes in four ways.

Original entry on oeis.org

13896, 1979, 1252, 626, 470, 256, 224, 225, 226, 227, 221, 222, 223, 203, 204, 205, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,3,4) in Alter's notation.

			a(3) = 13896 = 1^3 + 12^3 + 23^3 = 2^3 + 4^3 + 24^3 = 4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3.
a(4) = 1979 = 1^3 + 5^3 + 5^3 + 12^3 = 2^3 + 3^3 + 6^3 + 12^3 = 5^3 + 5^3 + 9^3 + 10^3 = 6^3 + 6^3 + 6^3 + 11^3.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 11, page 194.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343081, A343084, A343086.

LinearRecurrence[{2,-1},{13896,1979,1252,626,470,256,224,225,226,227,221,222,223,203,204,205,171,172},60] (* Harvey P. Dale, Aug 06 2022 *)

a(n) = n + 152 for n >= 19.

A343080 a(n) is the smallest number that is the sum of n positive squares in three ways.

Original entry on oeis.org

325, 54, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,2,3) in Alter's notation.

			a(2) = 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(3) = 54 = 1^2 + 2^2 + 7^2 = 2^2 + 5^2 + 5^2 = 3^2 + 3^2 + 6^2.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 6, page 191.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343083, A343081, A343082, A343083, A343084.

a(n) = n + 24 for n >= 4.

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A342902 a(n) is the smallest number that is the sum of n positive cubes in two ways.

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A343082 a(n) is the smallest number that is the sum of n positive 4th powers in three ways.

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A343083 a(n) is the smallest number that is the sum of n positive 5th powers in three ways.

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A343085 a(n) is the smallest number that is the sum of n positive cubes in four ways.

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A343080 a(n) is the smallest number that is the sum of n positive squares in three ways.

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