A350405 -id:A350405 - cp's OEIS frontend

A350397 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

305, 1980, 1900, 3321, 5256, 8310, 12516, 17108, 24832, 34249, 42381, 61697, 78766, 106956, 132994, 170325, 203415, 266595, 322943, 393828, 475520, 569416, 695799, 823447, 958300, 1149125, 1313545, 1565055, 1802736, 2088119, 2376250, 2748270, 3135195, 3548876

Offset: 3

Ilya Gutkovskiy, Dec 29 2021

nonn

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

Martin Ehrenstein, Table of n, a(n) for n = 3..52
Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A080851, A350210, A350405.

a(10)-a(22) from Michael S. Branicky, Dec 29 2021

a(23)-a(36) from Martin Ehrenstein, Jan 14 2022

A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

81, 1105, 205427, 483031, 9402323, 6232341, 79324200, 768459127, 2265692766, 2413112833, 6737406626, 150437989675, 45319359337, 15140186701

Offset: 3

Ilya Gutkovskiy, Jun 28 2024

			a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Eric Weisstein's World of Mathematics, Polygonal Number
Michael S. Branicky, Python program for A374144

Cf. A057145, A093195, A332989, A342326, A350405, A374141, A374142, A374143.

Python
```
# see linked program
```

a(9)-a(16) from Michael S. Branicky, Jun 30 2024

A350423 a(n) is the smallest n-gonal number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

190, 289, 330, 561

Offset: 3

Ilya Gutkovskiy, Dec 31 2021

			For n = 3: 190 = 1 + 36 + 153 = 15 + 55 + 120 = 21 + 78 + 91.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A350405.

A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

47, 142, 344, 652, 834, 1542, 2263, 3090, 4792, 4570, 5844, 8480, 9571, 10542, 15892, 18202, 19587, 23166, 26732, 32040, 36371, 39730, 44709, 52940, 55141, 60362, 67705, 79624, 86906, 89266, 103591, 116246, 126610, 131462, 135324, 148190, 158152, 162422, 186126, 200254

Offset: 3

Ilya Gutkovskiy, Jul 02 2024

nonn

			a(3) = 47 = 1 + 3 + 15 + 28 = 1 + 10 + 15 + 21 = 3 + 6 + 10 + 28.
a(4) = 142 = 1^2 + 2^2 + 4^2 + 11^2 = 1^2 + 4^2 + 5^2 + 10^2 = 2^2 + 5^2 + 7^2 + 8^2 = 3^2 + 4^2 + 6^2 + 9^2.

Michael S. Branicky, Table of n, a(n) for n = 3..204
Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A025417, A057145, A350405, A374144, A374273.

a(36) and beyond from Michael S. Branicky, Jul 08 2024

A374273 a(n) is the smallest number which can be represented as the sum of three distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

37, 161, 498, 1666, 2546, 7434, 16609, 25952, 48786, 49861, 72347, 127335, 183289, 196469, 416913, 466546, 494369, 506649, 801010, 1401011, 2372586, 1414009, 2003027, 3274986, 2927260, 2721677, 5592756, 8016592, 6632759, 7057914, 8401837, 13248146, 11648679, 8650006

Offset: 3

Ilya Gutkovskiy, Jul 02 2024

nonn

			a(3) = 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.
a(4) = 161 = 1^2 + 4^2 + 12^2 = 2^2 + 6^2 + 11^2 = 4^2 + 8^2 + 9^2 = 5^2 + 6^2 + 10^2.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A025415, A057145, A350405, A374144, A374274.

a(21) and beyond from Michael S. Branicky, Jul 08 2024

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

A363253 a(n) is the smallest n-gonal number which can be represented as the sum of distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

28, 121, 210

Offset: 3

Ilya Gutkovskiy, May 23 2023

a(8) = 736, a(9) = 969.

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

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A057145, A350207, A350405, A350423.

A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

Original entry on oeis.org

19, 90, 162, 299, 509, 816, 1248, 1837, 2619, 3634, 4926, 6543, 8537, 10964, 13884, 17361, 21463, 26262, 31834, 38259, 45621, 54008, 63512, 74229, 86259, 99706, 114678, 131287, 149649, 169884, 192116, 216473, 243087, 272094, 303634, 337851, 374893, 414912, 458064, 504509

Offset: 3

Ilya Gutkovskiy, Nov 05 2024

nonn

From David A. Corneth, Nov 06 2024: (Start)
a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:
A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).
We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.
This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)

			a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A006484, A025377, A057145, A350405, A374144, A374256, A374287.

From David A. Corneth, Nov 06 2024: (Start)
a(n) >= A006484(n).
Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)
Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.

a(12)-a(36) from Michael S. Branicky, Nov 06 2024

More terms from David A. Corneth, Nov 10 2024

cp's OEIS Frontend

A350397 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

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A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A350423 a(n) is the smallest n-gonal number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or -1 if none exists.

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A374274 a(n) is the smallest number which can be represented as the sum of four distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A374273 a(n) is the smallest number which can be represented as the sum of three distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number 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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A363253 a(n) is the smallest n-gonal number which can be represented as the sum of distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions