A343080 -id:A343080 - cp's OEIS frontend

A343081 a(n) is the smallest number that is the sum of n positive cubes in three or more ways.

5104, 1225, 766, 221, 222, 223, 224, 197, 163, 164, 165, 166, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177

Offset: 3

Sean A. Irvine, Apr 04 2021

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

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
a(4) = 1225 = 1^3 + 2^3 + 6^3 + 10^3 = 3^3 + 7^3 + 7^3 + 8^3 = 4^3 + 6^3 + 6^3 + 9^3.
a(9) = 224 = 6^3 + 8*1^3 = 3*4^3 + 3^3 + 5*1^3 = 5^3 + 4^3 + 4*2^3 + 3*1^3.

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 7, page 192.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A342902, A343080, A343082, A343083, A343085.

a(n) = n + 124 for n >= 15.

Corrected by Robert Israel, Apr 05 2021

a(9) reverted by Sean A. Irvine, Apr 18 2021

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.

A343084 a(n) is the smallest number that is the sum of n positive squares in four ways.

Original entry on oeis.org

1105, 129, 52, 53, 36, 37, 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

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,2,4) in Alter's notation.

			a(2) = 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.
a(3) = 129 = 1^2 + 8^2 + 8^2 = 2^2 + 2^2 + 11^2 = 2^2 + 5^2 + 10^2 = 4^2 + 7^2 + 8^2.

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 10, page 193.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A343080, A343085, A343086.

LinearRecurrence[{2,-1},{1105,129,52,53,36,37,32,33},70] (* Harvey P. Dale, Mar 18 2025 *)

a(n) = n + 24 for n >= 8.

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A343081 a(n) is the smallest number that is the sum of n positive cubes in three or more 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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A343084 a(n) is the smallest number that is the sum of n positive squares in four ways.

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