A319877 -id:A319877 - cp's OEIS frontend

A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202

Offset: 1

Gus Wiseman, Dec 17 2018

nonn

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}}. This sequence lists all MM-numbers of square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  26: {{},{1,2}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  51: {{1},{4}}
  53: {{1,1,1,1}}
  55: {{2},{3}}
  58: {{},{1,3}}
  59: {{7}}
  65: {{2},{1,2}}
  67: {{8}}
  69: {{1},{2,2}}
  74: {{},{1,1,2}}

Cf. A003963, A057151, A064573, A120732, A319616, A319877, A320325, A322527, A322530.

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

A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601

Offset: 1

Gus Wiseman, Dec 17 2018

nonn

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}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

			The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   97: {{3,3}}
  121: {{3},{3}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  185: {{2},{1,1,2}}
  195: {{1},{2},{1,2}}
  207: {{1},{1},{2,2}}
  225: {{1},{1},{2},{2}}
  227: {{4,4}}
  289: {{4},{4}}
  541: {{1,1,3,3}}
  661: {{5,5}}
  679: {{1,1},{3,3}}
  687: {{1},{1,3,3}}
  781: {{3},{1,1,3}}
  841: {{1,3},{1,3}}
  847: {{1,1},{3},{3}}
  873: {{1},{1},{3,3}}
  957: {{1},{3},{1,3}}
  961: {{5},{5}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

cp's OEIS Frontend

A319899 Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).

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