A350397 -id:A350397 - cp's OEIS frontend

A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

140, 490, 1055, 1872, 2610, 4255, 5011, 8708, 7497, 10819, 12860, 15636, 18055, 24275, 27373, 28146, 30826, 38178, 41849, 44025, 36165, 47621, 57896, 64648, 60064, 67125, 71975, 81820, 77701, 91584, 91320, 99835, 98916, 108686, 112606, 123180, 120919, 142270

Offset: 3

Ilya Gutkovskiy, Dec 19 2021

nonn

			For n = 3: 140 = 1 + 20 + 35 + 84 = 56 + 84 = 20 + 120. - _Martin Ehrenstein_, Jan 09 2022

Martin Ehrenstein, Table of n, a(n) for n = 3..1002
Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350205, A350206, A350207, A350209, A350397.

a(35)-a(40) from Martin Ehrenstein, Jan 09 2022

A360217 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero tetrahedral numbers in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 140, 305, 315, 435, 644, 830, 1141, 1425, 1925, 2380, 3010, 3805, 4720, 5806, 7095, 8510, 10200, 12020, 14115, 16460, 19131, 21990, 25425, 29275, 33495, 37425, 42680, 48300, 54545, 60711, 68391, 75726, 84815, 93370, 103250, 114115, 125360, 137831, 150995, 165545, 179830

Offset: 1

Ilya Gutkovskiy, Jan 30 2023

nonn

			For n = 3: 305 = 1 + 84 + 220 = 20 + 120 + 165 = 56 + 84 + 165.

Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292, A350205, A350288, A350397, A360218.

More terms from David A. Corneth, Jan 30 2023

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

A352975 a(n) is the smallest number which can be represented as the sum of n distinct centered n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

96, 192, 330, 504, 840, 1304, 1872, 2910, 3971, 5340, 6851, 8932, 11700, 14496, 18258, 22410, 27265, 32620, 39606, 47124, 55545, 65448, 76050, 87854, 101925, 116956, 134125, 152340, 173538, 195424, 220473, 246942, 276570, 306756, 340918, 377644, 418821, 462720

Offset: 3

Ilya Gutkovskiy, Apr 13 2022

nonn

If a(n) exists, then n divides a(n). - Thomas Scheuerle, Apr 13 2022

			For n = 3: 96 = 1 + 10 + 85 = 1 + 31 + 64 = 19 + 31 + 46.

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A101321, A350209, A350397, A350405.

a(n) >= n*binomial(n + 2, 3) + n, if a(n) exists. - Thomas Scheuerle, Apr 13 2022

a(10)-a(16) from Thomas Scheuerle, Apr 13 2022

a(17)-a(40) from Michael S. Branicky, May 19 2022

A359321 a(n) is the smallest n-gonal pyramidal number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

2300, 6201, 8125, 6391

Offset: 3

Ilya Gutkovskiy, Dec 25 2022

			For n = 3: 2300 = 1 + 969 + 1330 = 56 + 220 + 2024 = 165 + 364 + 1771.
For n = 4: 6201 = 1 + 91 + 1785 + 4324 = 1 + 285 + 1015 + 4900 = 30 + 140 + 506 + 5525 = 91 + 819 + 1496 + 3795.

Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350210, A350397, A350423.

cp's OEIS Frontend

A350210 a(n) is the smallest positive integer which can be represented as the sum of distinct nonzero n-gonal pyramidal numbers in exactly n ways, or 0 if no such integer exists.

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A360217 a(n) is the smallest positive integer which can be represented as the sum of n distinct nonzero tetrahedral numbers in exactly n ways, or -1 if no such integer 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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A352975 a(n) is the smallest number which can be represented as the sum of n distinct centered n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A359321 a(n) is the smallest n-gonal pyramidal number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if none exists.

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