OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with at most one odd prime index.
Also Heinz numbers of partitions with conjugate alternating sum <= 1.
EXAMPLE
The terms and their prime indices begin:
1: {} 23: {9} 49: {4,4}
2: {1} 26: {1,6} 51: {2,7}
3: {2} 27: {2,2,2} 53: {16}
5: {3} 29: {10} 54: {1,2,2,2}
6: {1,2} 31: {11} 57: {2,8}
7: {4} 33: {2,5} 58: {1,10}
9: {2,2} 35: {3,4} 59: {17}
11: {5} 37: {12} 61: {18}
13: {6} 38: {1,8} 63: {2,2,4}
14: {1,4} 39: {2,6} 65: {3,6}
15: {2,3} 41: {13} 67: {19}
17: {7} 42: {1,2,4} 69: {2,9}
18: {1,2,2} 43: {14} 71: {20}
19: {8} 45: {2,2,3} 73: {21}
21: {2,4} 47: {15} 74: {1,12}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Count[Reverse[primeMS[#]], _?OddQ]<=1&]
CROSSREFS
These are the positions of 0's and 1's in A257991.
The conjugate partitions are ranked by A349151.
A122111 is a representation of partition conjugation.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 10 2021
STATUS
approved