A350241 -id:A350241 - cp's OEIS frontend

A350405 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

37, 142, 285, 536, 911, 1268, 1909, 2713, 3876, 5179, 6891, 8901, 11190, 14384, 18087, 21697, 27055, 32166, 39111, 46560, 53892, 64412, 73949, 86778, 98202, 113635, 130088, 148051, 167505, 190968, 214955, 240143, 269775, 297615, 331201, 367429, 409179, 451340, 497830

Offset: 3

Ilya Gutkovskiy, Dec 29 2021

nonn

			For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.

David A. Corneth, Table of n, a(n) for n = 3..102
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A006484, A025443, A057145, A307598, A350207, A350241, A350288.

Do[i=1;While[b=PolygonalNumber[n,Range@i++];!IntegerQ[t=Min[First/@Select[Tally[Select[Total/@Subsets[b,{n}],#<=Max@b&]],Last@#==n&]]]];Print@t,{n,3,10}] (* Giorgos Kalogeropoulos, Dec 30 2021 *)

a(n) >= A006484(n). - David A. Corneth, Dec 30 2021

a(10)-a(31) from Michael S. Branicky, Dec 29 2021

More terms from David A. Corneth, Dec 30 2021

A350288 a(n) is the smallest number which can be represented as the sum of n distinct nonzero triangular numbers in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 16, 37, 64, 83, 128, 177, 204, 270, 352, 430, 533, 632, 764, 893, 1102, 1256, 1443, 1630, 1855, 2141, 2384, 2699, 3053, 3378, 3753, 4176, 4620, 5068, 5570, 6107, 6654, 7253, 7904, 8526, 9241, 9975, 10699, 11533, 12401, 13301, 14189, 15179, 16233, 17286, 18412

Offset: 1

Ilya Gutkovskiy, Dec 23 2021

nonn

			For n = 2: 16 = 1 + 15 = 6 + 10.
For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.

Cf. A000217, A265134, A307597, A307598, A341021-A341027, A342326, A350241.

More terms from Jinyuan Wang, Dec 26 2021

A374287 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 65, 62, 90, 103, 136, 200, 276, 376, 481, 625, 806, 975, 1183, 1415, 1688, 1989, 2325, 2698, 3110, 3563, 4059, 4600, 5188, 5825, 6513, 7254, 8050, 8903, 9815, 10788, 11824, 12925, 14093, 15330, 16638, 18019, 19475, 21008, 22620, 24313, 26089, 27950, 29898

Offset: 1

Ilya Gutkovskiy, Jul 02 2024

sign

			a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
a(3) = 62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2.

Index entries for sequences related to sums of squares

Cf. A000330, A093195, A350241.

a(12) and beyond from Michael S. Branicky, Jul 02 2024

A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 1729, 5104, 4445, 4509, 4662, 5454, 6210, 9045, 11124, 14967, 17964, 22051, 26209, 32697, 39564, 46908, 56070, 66222, 78912, 92961, 105841, 123732, 143200, 162801, 188154, 212220, 241614, 271405, 307448, 344016, 383607, 428624, 475273, 529830, 586664, 645120

Offset: 1

Ilya Gutkovskiy, Dec 22 2021

nonn

			For n = 2: 1729 = 1^3 + 12^3 = 9^3 + 10^3.
For n = 3: 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Index entries for sequences related to sums of cubes

Cf. A000578, A025401, A025419, A025421, A275154, A350241.

a(16)-a(27) from Michael S. Branicky, Dec 22 2021

More terms from Jinyuan Wang, Dec 30 2021

A360218 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero square pyramidal numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 5580, 2814, 1980, 1595, 1700, 2175, 2415, 2830, 3740, 4810, 5995, 7610, 9240, 11380, 13896, 16506, 19735, 23150, 27441, 32085, 36721, 42755, 49570, 56610, 65135, 73165, 83021, 93835, 105671, 118255, 132545, 147546, 163516, 182155, 201040, 222371, 244280, 267856

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

			For n = 3: 2814 = 14 + 1015 + 1785 = 55 + 650 + 2109 = 140 + 204 + 2470.

Michael S. Branicky, Table of n, a(n) for n = 1..80
Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A000330, A350206, A350241, A350397, A360217.

a(18)-a(33) from Michael S. Branicky, Feb 04 2023

a(34) and beyond from Michael S. Branicky, Feb 18 2023

A360306 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero fourth powers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 635318657, 811538, 300834, 185299, 138595, 143651, 154292, 197748, 225733, 291269, 374790, 474071, 586056, 715192, 857513, 1057689, 1330554, 1602250, 1919146, 2329547, 2786843, 3246204, 3899260, 4642700, 5378141, 6377822, 7342638, 8527103, 9787839, 11241455, 12978656

Offset: 1

Ilya Gutkovskiy, Feb 02 2023

nonn

			For n = 2: 635318657 = 59^4 + 158^4 = 133^4 + 134^4.
For n = 3: 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.
For n = 4: 300834 = 1^4 + 4^4 + 12^4 + 23^4 = 1^4 + 16^4 + 18^4 + 19^4 = 3^4 + 6^4 + 18^4 + 21^4 = 7^4 + 14^4 + 16^4 + 21^4.

Bert Dobbelaere, Table of n, a(n) for n = 1..79
Bert Dobbelaere, C++ program
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A146756, A350241, A350270.

a(18)-a(19) from Michael S. Branicky, Feb 04 2023

More terms from Bert Dobbelaere, Feb 11 2023

A375335 a(n) is the smallest positive integer whose square can be represented as the sum of n distinct nonzero squares in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 1, 25, 23, 17

Offset: 0

Ilya Gutkovskiy, Aug 12 2024

a(6) = 16, a(10) = 25; a(5) > 500, a(7..9) > 100 if not -1 - Michael S. Branicky, Aug 13 2024

			a(2) = 25: 25^2 =  7^2 + 24^2
                = 15^2 + 20^2.
.
a(3) = 23: 23^2 = 3^2 +  6^2 + 22^2
                = 3^2 + 14^2 + 18^2
                = 6^2 + 13^2 + 18^2.
.
a(4) = 17: 17^2 = 2^2 + 4^2 + 10^2 + 13^2
                = 2^2 + 5^2 +  8^2 + 14^2
                = 2^2 + 8^2 + 10^2 + 11^2
                = 3^2 + 6^2 + 10^2 + 12^2.

Index entries for sequences related to sums of squares

Cf. A006339, A374805, A350241.

cp's OEIS Frontend

A350405 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

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A350288 a(n) is the smallest number which can be represented as the sum of n distinct nonzero triangular numbers in exactly n ways, or 0 if no such number exists.

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A374287 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly 2 ways, or -1 if no such number exists.

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A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

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A360218 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero square pyramidal numbers in exactly n ways, or -1 if no such integer exists.

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A360306 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero fourth powers in exactly n ways, or -1 if no such integer exists.

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A375335 a(n) is the smallest positive integer whose square can be represented as the sum of n distinct nonzero squares in exactly n ways, or -1 if no such integer exists.

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