A374144 -id:A374144 - cp's OEIS frontend

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.

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

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

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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