OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 19: {8} 48: {1,1,1,1,2}
2: {1} 21: {2,4} 49: {4,4}
3: {2} 23: {9} 53: {16}
4: {1,1} 24: {1,1,1,2} 54: {1,2,2,2}
5: {3} 25: {3,3} 59: {17}
6: {1,2} 27: {2,2,2} 61: {18}
7: {4} 29: {10} 63: {2,2,4}
8: {1,1,1} 31: {11} 64: {1,1,1,1,1,1}
9: {2,2} 32: {1,1,1,1,1} 65: {3,6}
11: {5} 36: {1,1,2,2} 67: {19}
12: {1,1,2} 37: {12} 71: {20}
13: {6} 41: {13} 72: {1,1,1,2,2}
16: {1,1,1,1} 42: {1,2,4} 73: {21}
17: {7} 43: {14} 79: {22}
18: {1,2,2} 47: {15} 81: {2,2,2,2}
MATHEMATICA
Select[Range[100], With[{y=PrimePi/@First/@FactorInteger[#]}, And@@Table[y[[i]]==y[[i-1]]||y[[i]]==2*y[[i-1]], {i, 2, Length[y]}]]&]
CROSSREFS
The first condition alone gives A000961 (perfect powers).
The second condition alone is counted by A154402.
These partitions are counted by A342337.
A018819 counts partitions into powers of 2.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A045690 counts sets with maximum n in with adjacent elements y < 2x.
A274199 counts compositions with adjacent parts x < 2y.
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
A342342 counts strict compositions with adjacent parts x <= 2y and y <= 2x.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 11 2021
STATUS
approved