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A325183
Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.
9
1, 2, 3, 3, 5, 6, 7, 5, 10, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 14, 21, 22, 23, 14, 21, 26, 21, 22, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 22, 41, 42, 43, 26, 42, 46, 47, 22, 55, 42, 51, 34, 53, 35, 55, 26, 57, 58, 59, 42, 61, 62, 66, 13, 65, 66, 67
OFFSET
1,2
COMMENTS
The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
LINKS
Eric Weisstein's World of Mathematics, Graph Distance.
EXAMPLE
The partition with Heinz number 7865 is (6,5,5,3), with diagram
o o o o o o
o o o o o
o o o o o
o o o
with origin-to-boundary graph-distances
4 4 4 3 2 1
3 3 3 2 1
2 2 2 1 1
1 1 1
giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1, #]&)/@Sort[ptn, Greater], {Length[ptn], Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat, Select[Position[mat, 1], Times@@Extract[mat, {#+{1, 0}, #+{0, 1}}]==0&]->0];
Table[Times@@Prime/@If[n==1, {}, -Differences[Map[Total, Drop[FixedPointList[corpos, ptnmat[primeptn[n]]], -1], 2]]], {n, 30}]
CROSSREFS
The only terms appearing only once are the primorials A002110.
The union consists of all squarefree numbers A005117.
Sequence in context: A155918 A344705 A331170 * A343247 A097248 A097247
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 08 2019
STATUS
approved