login
A325196
Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.
9
3, 4, 9, 10, 12, 15, 18, 20, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 126, 140, 150, 294, 315, 330, 350, 420, 441, 462, 490, 495, 525, 550, 588, 630, 660, 693, 700, 735, 770, 825, 882, 924, 980, 990, 1050, 1100, 1155, 1386, 1470, 1540, 1650, 2730, 3234
OFFSET
1,1
COMMENTS
The enumeration of these partitions by sum is given by A325191.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2}
4: {1,1}
9: {2,2}
10: {1,3}
12: {1,1,2}
15: {2,3}
18: {1,2,2}
20: {1,1,3}
42: {1,2,4}
45: {2,2,3}
50: {1,3,3}
60: {1,1,2,3}
63: {2,2,4}
70: {1,3,4}
75: {2,3,3}
84: {1,1,2,4}
90: {1,2,2,3}
100: {1,1,3,3}
105: {2,3,4}
126: {1,2,2,4}
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]];
Select[Range[1000], otbmax[primeptn[#]]-otb[primeptn[#]]==1&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 11 2019
STATUS
approved