|
|
A349150
|
|
Heinz numbers of integer partitions with at most one odd part.
|
|
5
|
|
|
1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 18, 19, 21, 23, 26, 27, 29, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 74, 77, 78, 79, 81, 83, 86, 87, 89, 91, 93, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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.
A300063 ranks partitions of odd numbers, counted by A058695 up to 0's.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325698 ranks partitions with as many even as odd parts, counted by A045931.
A345958 ranks partitions with alternating sum 1.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf. A000290, A000700, A001222, A027187, A027193, A028260, A035363, A047993, A215366, A257992, A277579, A326841.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|