login
A321650
Irregular triangle whose n-th row is the reversed conjugate of the integer partition with Heinz number n.
14
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, 1
OFFSET
1,4
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
FORMULA
a(n,i) = A112798(A122111(n),i).
EXAMPLE
Triangle begins:
1
1 1
2
1 1 1
1 2
1 1 1 1
3
2 2
1 1 2
1 1 1 1 1
1 3
1 1 1 1 1 1
1 1 1 2
1 2 2
4
1 1 1 1 1 1 1
2 3
1 1 1 1 1 1 1 1
1 1 3
1 1 2 2
1 1 1 1 2
1 1 1 1 1 1 1 1 1
The sequence of reversed dual partitions begins: (), (1), (11), (2), (111), (12), (1111), (3), (22), (112), (11111), (13), (111111), (1112), (122), (4), (1111111), (23), (11111111), (113), (1122), (11112), (111111111), (14), (222), (111112), (33), (1113), (1111111111), (123).
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Sort[conj[primeMS[n]]], {n, 50}]
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Nov 15 2018
STATUS
approved