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A356940
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MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).
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6
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1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 96, 104, 108, 113, 117, 128, 144, 156, 162, 169, 192, 208, 216, 226, 234, 243, 256, 288, 312, 324, 338, 339, 351, 384, 416, 432, 452, 468, 486, 507, 512, 576, 624, 648
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OFFSET
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1,2
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COMMENTS
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An initial interval is a set {1,2,...,n} for some n >= 0.
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}}.
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LINKS
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EXAMPLE
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The initial terms and corresponding multisets of multisets:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
6: {{},{1}}
8: {{},{},{}}
9: {{1},{1}}
12: {{},{},{1}}
13: {{1,2}}
16: {{},{},{},{}}
18: {{},{1},{1}}
24: {{},{},{},{1}}
26: {{},{1,2}}
27: {{1},{1},{1}}
32: {{},{},{},{},{}}
36: {{},{},{1},{1}}
39: {{1},{1,2}}
48: {{},{},{},{},{1}}
52: {{},{},{1,2}}
54: {{},{1},{1},{1}}
64: {{},{},{},{},{},{}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
chinQ[y_]:=y==Range[Length[y]];
Select[Range[100], And@@chinQ/@primeMS/@primeMS[#]&]
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CROSSREFS
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This is the initial version of A356939.
A000688 counts factorizations into prime powers.
A001222 counts prime factors with multiplicity.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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