|
|
A356930
|
|
Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.
|
|
6
|
|
|
1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 27, 28, 29, 31, 32, 33, 36, 38, 42, 44, 48, 49, 53, 54, 56, 57, 58, 59, 62, 63, 64, 66, 71, 72, 76, 77, 79, 81, 83, 84, 87, 88, 93, 96, 97, 98, 99, 106, 108, 112, 114, 116, 118, 121, 124, 126, 127
(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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The combined size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
|
|
LINKS
|
|
|
EXAMPLE
|
The initial terms and corresponding multisets of multisets:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
6: {{},{1}}
7: {{1,1}}
8: {{},{},{}}
9: {{1},{1}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
16: {{},{},{},{}}
18: {{},{1},{1}}
19: {{1,1,1}}
21: {{1},{1,1}}
22: {{},{3}}
24: {{},{},{},{1}}
27: {{1},{1},{1}}
28: {{},{},{1,1}}
29: {{1,3}}
31: {{5}}
32: {{},{},{},{},{}}
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], And@@(OddQ[Times@@primeMS[#]]&/@primeMS[#])&]
|
|
CROSSREFS
|
Factorizations of this type are counted by A356931.
The version for odd lengths instead of parts is A356935, ranked by A089259.
A000688 counts factorizations into prime powers.
A001222 counts prime factors with multiplicity.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|