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 A320057 Heinz numbers of spanning sum-product knapsack partitions. 7
 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 101, 103, 105 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A spanning sum-product knapsack partition is a finite multiset m of positive integers such that every sum of products of parts of any multiset partition of m is distinct. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). Differs from A320058 in having 1155, 1625, 1815, 1875, 1911, ... and lacking 20, 28, 42, 44, 52, ... LINKS EXAMPLE The sequence of all spanning sum-product knapsack partitions begins: (), (1), (2), (1,1), (3), (2,1), (4), (1,1,1), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (3,3), (6,1). A complete list of sums of products of multiset partitions of the partition (5,4,3,2) is:         (2*3*4*5) = 120       (2)+(3*4*5) = 62       (3)+(2*4*5) = 43       (4)+(2*3*5) = 34       (5)+(2*3*4) = 29       (2*3)+(4*5) = 26       (2*4)+(3*5) = 23       (2*5)+(3*4) = 22     (2)+(3)+(4*5) = 25     (2)+(4)+(3*5) = 21     (2)+(5)+(3*4) = 19     (3)+(4)+(2*5) = 17     (3)+(5)+(2*4) = 16     (4)+(5)+(2*3) = 15   (2)+(3)+(4)+(5) = 14 These are all distinct, and the Heinz number of (5,4,3,2) is 1155, so 1155 belongs to the sequence. MATHEMATICA multWt[n_]:=If[n==1, 1, Times@@Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]^k]]; facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; Select[Range[100], UnsameQ@@Table[Plus@@multWt/@f, {f, facs[#]}]&] CROSSREFS Cf. A001970, A056239, A066739, A108917, A112798, A292886, A299702, A301899, A318949, A319318, A319913. Cf. A267597, A320052, A320053, A320054, A320055, A320056, A320058. Sequence in context: A317717 A181709 A320058 * A033106 A119485 A058363 Adjacent sequences:  A320054 A320055 A320056 * A320058 A320059 A320060 KEYWORD nonn AUTHOR Gus Wiseman, Oct 04 2018 STATUS approved

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Last modified January 21 16:54 EST 2020. Contains 331114 sequences. (Running on oeis4.)