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
Examples
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.
Links
- 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.
Programs
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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}]