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 A319855 Minimum number that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains. 4
 0, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 5, 1, 7, 4, 8, 3, 6, 5, 9, 2, 6, 6, 6, 4, 10, 5, 11, 1, 7, 7, 7, 4, 12, 8, 8, 3, 13, 6, 14, 5, 7, 9, 15, 2, 8, 6, 9, 6, 16, 6, 8, 4, 10, 10, 17, 5, 18, 11, 8, 1, 9, 7, 19, 7, 11, 7, 20, 4, 21, 12, 8, 8, 9, 8, 22, 3, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). LINKS FORMULA a(1) = 0, a(n) = max(A056239(n) - A007814(n), 1). - Charlie Neder, Oct 03 2018 EXAMPLE a(30) = 5 because the minimum number that can be obtained starting with (3,2,1) is 3+2*1 = 5. MATHEMATICA ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]]; nexos[ptn_]:=If[Length[ptn]==0, {0}, Union@@Select[ReplaceListRepeated[{Sort[ptn]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]]; Table[Min[nexos[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {n, 100}] CROSSREFS Cf. A000792, A001970, A048249, A056239, A066739, A066815, A070960, A201163, A319850, A318948, A318949, A319841, A319856. Sequence in context: A167430 A056892 A136523 * A228731 A163507 A003963 Adjacent sequences:  A319852 A319853 A319854 * A319856 A319857 A319858 KEYWORD nonn AUTHOR Gus Wiseman, Sep 29 2018 STATUS approved

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Last modified July 26 20:17 EDT 2021. Contains 346294 sequences. (Running on oeis4.)