A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 41, 43, 47, 49, 51, 53, 55, 59, 64, 67, 73, 79, 81, 83, 85, 93, 97, 101, 103, 109, 113, 121, 123, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 169, 177, 179

Offset

1,2

Comments

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform and regular, so its MM-number 15463 belongs to the sequence.

Links

Table of n, a(n) for n=1..61.

Example

The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:
   1: {}                   33: {{1},{3}}            109: {{10}}
   2: {{}}                 41: {{6}}                113: {{1,2,3}}
   3: {{1}}                43: {{1,4}}              121: {{3},{3}}
   4: {{},{}}              47: {{2,3}}              123: {{1},{6}}
   5: {{2}}                49: {{1,1},{1,1}}        125: {{2},{2},{2}}
   7: {{1,1}}              51: {{1},{4}}            127: {{11}}
   8: {{},{},{}}           53: {{1,1,1,1}}          128: {{},{},{},{},{},{}}
   9: {{1},{1}}            55: {{2},{3}}            131: {{1,1,1,1,1}}
  11: {{3}}                59: {{7}}                137: {{2,5}}
  13: {{1,2}}              64: {{},{},{},{},{},{}}  139: {{1,7}}
  15: {{1},{2}}            67: {{8}}                149: {{3,4}}
  16: {{},{},{},{}}        73: {{2,4}}              151: {{1,1,2,2}}
  17: {{4}}                79: {{1,5}}              155: {{2},{5}}
  19: {{1,1,1}}            81: {{1},{1},{1},{1}}    157: {{12}}
  23: {{2,2}}              83: {{9}}                161: {{1,1},{2,2}}
  25: {{2},{2}}            85: {{2},{4}}            163: {{1,8}}
  27: {{1},{1},{1}}        93: {{1},{5}}            165: {{1},{2},{3}}
  29: {{1,3}}              97: {{3,3}}              167: {{2,6}}
  31: {{5}}               101: {{1,6}}              169: {{1,2},{1,2}}
  32: {{},{},{},{},{}}    103: {{2,2,2}}            177: {{1},{7}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], And[SameQ@@PrimeOmega/@primeMS[#], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

Crossrefs

Cf. A005176, A007016, A112798, A271103, A283877, A299353, A302242, A306017, A319056, A319189, A320324, A321699, A321717, A322554, A322703, A322833.

Sequence in context: A316476 A056867 A320324 * A325394 A358378 A062491

Adjacent sequences: A321695 A321696 A321697 * A321699 A321700 A321701

Keyword

nonn

Author

Gus Wiseman, Dec 27 2018

Status

approved