A360217 -id:A360217 - cp's OEIS frontend

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.

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

A360431 a(n) is the smallest positive integer which can be represented as the sum of n distinct binomial coefficients binomial(k,n) for some k >= n in exactly n ways, or -1 if no such integer exists.

Original entry on oeis.org

1, 16, 305, 4396, 43093, 332193, 87172020, 273879343

Offset: 1

Ilya Gutkovskiy, Feb 07 2023

			For n = 2: 16 = C(2,2) + C(6,2) = C(4,2) + C(5,2) = 1 + 15 = 6 + 10.
For n = 3: 305 = C(3,3) + C(9,3) + C(12,3) = C(6,3) + C(10,3) + C(11,3) = C(8,3) + C(9,3) + C(11,3) = 1 + 84 + 220 = 20 + 120 + 165 = 56 + 84 + 165.
For n = 4: 4396 = C(5,4) + C(11,4) + C(14,4) + C(18,4) = C(8,4) + C(9,4) + C(16,4) + C(17,4) = C(9,4) + C(12,4) + C(13,4) + C(18,4) = C(10,4) + C(14,4) + C(15,4) + C(16,4) = 5 + 330 + 1001 + 3060 = 70 + 126 + 1820 + 2380 = 126 + 495 + 715 + 3060 = 210 + 1001 + 1365 + 1820.

Eric Weisstein's World of Mathematics, Binomial Coefficient

Cf. A007318, A350288, A360217.

a(8) from Michael S. Branicky, Feb 17 2023

cp's OEIS Frontend

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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