A317254 a(n) is the smallest integer such that for all s >= a(n), there are at least n-1 different partitions of s into n parts, namely {x_{11},x_{12},...,x_{1n}}, {x_{21},x_{22},...,x_{2n}},..., and {x_{n-1,1},x_{n-1,2},...,x_{n-1,n}}, such that the products of every set are equal.

19, 23, 23, 26, 27, 29, 31, 32, 35, 36, 38, 40, 42, 44, 45, 47, 49, 50, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 96, 97, 99, 100, 101

Offset

3,1

Links

Table of n, a(n) for n=3..60.

Byungchul Cha et al., An Investigation on Partitions with Equal Products, arXiv:1811.07451 [math.NT], 2018.

John B. Kelly, Partitions with equal products, Proc. Amer. Math. Soc. 15 (1964), 987-990.

Example

a(3)=19. From s=19 onward, there are at least 2 different partitions of s into 3 parts with equal products:
s=19: {12,4,3} & {9,8,2}:
  12 + 4 + 3 = 9 + 8 + 2 =  19;
  12 * 4 * 3 = 9 * 8 * 2 = 144;
s=20: {15,3,2} & {10,9,1}:
  15 + 3 + 2 = 10 + 9 + 1 = 20;
  15 * 3 * 2 = 10 * 9 * 1 = 90;
s=21: {16,3,2} & {12,8,1}:
  16 + 3 + 2 = 12 + 8 + 1 = 21;
  16 * 3 * 2 = 12 * 8 * 1 = 96.

Mathematica

Do[maxsumnotwork = 0;  Do[intpart = IntegerPartitions[sum, {n}];   prod = Table[Times @@ intpart[[i]], {i, Length[intpart]}];   prodtally = Tally[prod];   repeatprod = Select[prodtally, #[[2]] >= n - 1 &];   If[repeatprod == {}, maxsumnotwork = sum], {sum, 12, 200}];  Print[n, " ", maxsumnotwork + 1], {n, 3, 60}]

Crossrefs

Cf. A060277, A316945, A316946.

Sequence in context: A072305 A095222 A343740 * A226687 A070299 A159021

Adjacent sequences: A317251 A317252 A317253 * A317255 A317256 A317257

Keyword

nonn,more,hard

Author

Byungchul Cha, Adam Claman, Joshua Harrington, Ziyu Liu, Barbara Maldonado, Alexander M. Miller, Ann Palma, Wing Hong Tony Wong, Hongkwon V. Yi, Jul 25 2018

Status

approved