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A325186
Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.
4
3, 4, 15, 18, 21, 25, 27, 33, 36, 39, 51, 57, 69, 72, 87, 93, 105, 111, 123, 129, 141, 144, 147, 150, 159, 165, 175, 177, 183, 195, 201, 213, 219, 225, 231, 237, 245, 249, 250, 255, 267, 273, 275, 285, 288, 291, 300, 303, 309, 321, 325, 327, 339, 343, 345, 357
OFFSET
1,1
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 sequence of terms together with their prime indices begins:
3: {2}
4: {1,1}
15: {2,3}
18: {1,2,2}
21: {2,4}
25: {3,3}
27: {2,2,2}
33: {2,5}
36: {1,1,2,2}
39: {2,6}
51: {2,7}
57: {2,8}
69: {2,9}
72: {1,1,1,2,2}
87: {2,10}
93: {2,11}
105: {2,3,4}
111: {2,12}
123: {2,13}
129: {2,14}
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];
Select[Range[100], Apply[Plus, If[#==1, {}, FixedPointList[corpos, ptnmat[primeptn[#]]][[-3]]], {0, 1}]==2&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 08 2019
STATUS
approved