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A325040
Heinz numbers of integer partitions with the same product of parts as their conjugate.
16
1, 2, 6, 9, 20, 30, 49, 56, 70, 75, 81, 84, 90, 125, 176, 182, 210, 264, 350, 416, 441, 532, 540, 546, 624, 660, 735, 910, 1088, 1100, 1260, 1378, 1386, 1443, 1520, 1560, 1624, 1632, 1715, 2100, 2310, 2401, 2405, 2432, 2489, 2600, 3024, 3267, 3276, 3648, 3744
OFFSET
1,2
COMMENTS
For example, 182 is the Heinz number of (6,4,1) with product 24 and conjugate (3,2,2,2,1,1) with product also 24.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k).
The enumeration of these partitions by sum is given by A325039.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
6: {1,2}
9: {2,2}
20: {1,1,3}
30: {1,2,3}
49: {4,4}
56: {1,1,1,4}
70: {1,3,4}
75: {2,3,3}
81: {2,2,2,2}
84: {1,1,2,4}
90: {1,2,2,3}
125: {3,3,3}
176: {1,1,1,1,5}
182: {1,4,6}
210: {1,2,3,4}
264: {1,1,1,2,5}
350: {1,3,3,4}
416: {1,1,1,1,1,6}
MATHEMATICA
priptn[n_]:=If[n==1, {}, Reverse[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]}]];
Select[Range[100], Times@@priptn[#]==Times@@conj[priptn[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 25 2019
STATUS
approved