A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644

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. 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 multiset multisystems (meaning all vertex-degrees are 2).

Links

Table of n, a(n) for n=1..56.

Example

The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   14: {{},{1,1}}
   18: {{},{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   28: {{},{},{1,1}}
   36: {{},{},{1},{1}}
   46: {{},{2,2}}
   50: {{},{2},{2}}
   56: {{},{},{},{1,1}}
   72: {{},{},{},{1},{1}}
   92: {{},{},{2,2}}
   97: {{3,3}}
  100: {{},{},{2},{2}}
  112: {{},{},{},{},{1,1}}
  121: {{3},{3}}
  144: {{},{},{},{},{1},{1}}
  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}}
  184: {{},{},{},{2,2}}
  185: {{2},{1,1,2}}
  194: {{},{3,3}}
  195: {{1},{2},{1,2}}
  200: {{},{},{},{2},{2}}

Mathematica

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

Crossrefs

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

Sequence in context: A332514 A353168 A305214 * A020939 A282758 A073457

Adjacent sequences: A319874 A319875 A319876 * A319878 A319879 A319880

Keyword

nonn

Author

Gus Wiseman, Dec 17 2018

Status

approved