A343082 -id:A343082 - cp's OEIS frontend

A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

635318657, 2673, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292

Offset: 2

Sean A. Irvine, Apr 04 2021

nonn

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

			a(2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
a(3) = 2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4.

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 3, page 190.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003824, A046881, A342902, A343078, A343079, A343082, A343086.

a(n) = n + 240 for n >= 16.

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

Original entry on oeis.org

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.

A343086 a(n) is the smallest number that is the sum of n positive 4th powers in four ways.

Original entry on oeis.org

5978882, 236674, 20995, 6626, 2925, 2925, 2854, 1620, 1621, 777, 778, 779, 780, 781, 782, 528, 529, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

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

			a(3) = 5978882 = 3^4 + 40^4 + 43^4 = 8^4 + 37^4 + 45^4 = 15^4 + 32^4 + 47^4 = 23^4 + 25^4 + 48^4.
a(4) = 236674 = 1^4 + 2^4 + 7^4 + 22^4 = 3^4 + 6^4 + 18^4 + 19^4 = 7^4 + 14^4 + 16^4 + 19^4 = 8^4 + 16^4 + 17^4 + 17^4.

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 12, page 194.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A085559, A123075, A342897, A343077, A343082, A343084, A343085.

a(n) = n + 480 for n >= 32.

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.

cp's OEIS Frontend

A343077 a(n) is the smallest number that is the sum of n positive 4th powers in two ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A343086 a(n) is the smallest number that is the sum of n positive 4th powers in four ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula