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

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

1,2

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 (all vertex-degrees are 2) multiset partitions (no empty parts).

Links

Table of n, a(n) for n=1..49.

Example

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}}

Mathematica

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[#]]]&]

Crossrefs

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

Sequence in context: A257332 A350540 A179788 * A083582 A229500 A303188

Adjacent sequences: A319875 A319876 A319877 * A319879 A319880 A319881

Keyword

nonn

Author

Gus Wiseman, Dec 17 2018

Status

approved