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A356935
Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.
10
1, 3, 5, 9, 11, 15, 17, 19, 25, 27, 31, 33, 37, 41, 45, 51, 55, 57, 59, 61, 67, 71, 75, 81, 83, 85, 93, 95, 99, 103, 107, 109, 111, 113, 121, 123, 125, 127, 131, 135, 153, 155, 157, 165, 171, 177, 179, 181, 183, 185, 187, 191, 193, 197, 201, 205, 209, 211, 213
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 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}}.
EXAMPLE
The initial terms and corresponding multiset partitions:
1: {}
3: {{1}}
5: {{2}}
9: {{1},{1}}
11: {{3}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
33: {{1},{3}}
37: {{1,1,2}}
41: {{6}}
45: {{1},{1},{2}}
51: {{1},{4}}
55: {{2},{3}}
57: {{1},{1,1,1}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[Times@@Length/@primeMS/@primeMS[#]]&]
CROSSREFS
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Odd-size multisets are ctd by A000302, A027193, A058695, rkd by A026424.
Other types: A050330, A356932, A356933, A356934.
Sequence in context: A359408 A284310 A109324 * A190844 A191207 A285519
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 12 2022
STATUS
approved