|
|
A357861
|
|
Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.
|
|
7
|
|
|
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 40, 41, 43, 45, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
|
|
LINKS
|
|
|
EXAMPLE
|
The terms together with their prime indices begin:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
12: {1,1,2}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], GreaterEqual@@Total/@Split[primeMS[#]]&]
|
|
CROSSREFS
|
These partitions are counted by A304406.
These are the indices of rows in A354584 that are weakly decreasing.
The opposite (weakly increasing) version is A357875, counted by A304405.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|