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A319907 Number of distinct integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with the integer partition with Heinz number n. 0
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 1, 2, 4, 1, 4, 1, 2, 4, 1, 1, 4, 2, 3, 2, 2, 1, 5, 2, 4, 2, 1, 1, 5, 1, 1, 4, 4, 2, 4, 1, 2, 2, 4, 1, 5, 1, 1, 6, 2, 2, 4, 1, 5, 4, 1, 1, 7, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,15

COMMENTS

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

The Heinz number of (3,3,2) is 75 and we have

    3+3+2 = 8,

    3+3*2 = 9,

    3*3+2 = 11,

  (3+3)*2 = 12,

  3*(3+2) = 15,

    3*3*2 = 18,

so a(75) = 6.

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

mexos[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_?(#>1&), mie___, y_?(#>1&), afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]];

Table[Length[mexos[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {n, 100}]

CROSSREFS

Cf. A000792, A001970, A005520, A048249, A066739, A070960, A201163, A275870, A319850, A318949, A319855, A319856, A319909, A319912, A319913.

Sequence in context: A109374 A079706 A250005 * A078703 A090629 A248623

Adjacent sequences:  A319904 A319905 A319906 * A319908 A319909 A319910

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 01 2018

STATUS

approved

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Last modified April 16 10:51 EDT 2021. Contains 343037 sequences. (Running on oeis4.)