A350288 -id:A350288 - 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

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

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

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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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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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.

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