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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. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 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. 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: A303299 A072305 A095222 * A226687 A070299 A159021 Adjacent sequences:  A317251 A317252 A317253 * A317255 A317256 A317257 KEYWORD nonn,more,hard AUTHOR STATUS approved

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Last modified June 16 04:44 EDT 2021. Contains 345056 sequences. (Running on oeis4.)