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A325195
Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.
11
0, 0, 1, 1, 2, 0, 3, 2, 1, 1, 4, 1, 5, 2, 1, 3, 6, 1, 7, 1, 2, 3, 8, 2, 2, 4, 2, 2, 9, 0, 10, 4, 3, 5, 2, 2, 11, 6, 4, 2, 12, 1, 13, 3, 1, 7, 14, 3, 3, 1, 5, 4, 15, 2, 3, 2, 6, 8, 16, 1, 17, 9, 1, 5, 4, 2, 18, 5, 7, 1, 19, 3, 20, 10, 1, 6, 3, 3, 21, 3, 3, 11
OFFSET
1,5
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The partition (3,3) has Heinz number 25 and diagram
o o o
o o o
containing maximal triangular partition
o o
o
and contained in minimal triangular partition
o o o o
o o o
o o
o
so a(25) = 4 - 2 = 2.
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved