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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.

%I A317254 #16 Feb 24 2023 03:09:11
%S A317254 19,23,23,26,27,29,31,32,35,36,38,40,42,44,45,47,49,50,52,53,54,55,57,
%T A317254 58,59,61,62,63,64,66,67,69,70,71,73,74,75,77,78,80,81,82,84,85,86,87,
%U A317254 88,89,90,91,93,94,95,96,97,99,100,101
%N 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.
%H A317254 Byungchul Cha et al., <a href="https://arxiv.org/abs/1811.07451">An Investigation on Partitions with Equal Products</a>, arXiv:1811.07451 [math.NT], 2018.
%H A317254 John B. Kelly, <a href="http://dx.doi.org/10.1090/S0002-9939-1964-0168542-2">Partitions with equal products</a>, Proc. Amer. Math. Soc. 15 (1964), 987-990.
%e A317254 a(3)=19. From s=19 onward, there are at least 2 different partitions of s into 3 parts with equal products:
%e A317254 s=19: {12,4,3} & {9,8,2}:
%e A317254   12 + 4 + 3 = 9 + 8 + 2 =  19;
%e A317254   12 * 4 * 3 = 9 * 8 * 2 = 144;
%e A317254 s=20: {15,3,2} & {10,9,1}:
%e A317254   15 + 3 + 2 = 10 + 9 + 1 = 20;
%e A317254   15 * 3 * 2 = 10 * 9 * 1 = 90;
%e A317254 s=21: {16,3,2} & {12,8,1}:
%e A317254   16 + 3 + 2 = 12 + 8 + 1 = 21;
%e A317254   16 * 3 * 2 = 12 * 8 * 1 = 96.
%t A317254 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}]
%Y A317254 Cf. A060277, A316945, A316946.
%K A317254 nonn,more,hard
%O A317254 3,1
%A A317254 _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