A319272 Numbers whose prime multiplicities are distinct and whose prime indices are term of the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 31, 32, 37, 40, 44, 45, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 75, 76, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 103, 107, 108, 112, 121, 124, 125, 127, 128, 131, 135, 136, 144, 147, 148

Offset: 1

Author

Gus Wiseman, Sep 16 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n.

Also Matula-Goebel numbers of rooted trees in which the multiplicities in the multiset of branches directly under any given node are distinct.

Examples

			36 is not in the sequence because 36 = 2^2 * 3^2 does not have distinct prime multiplicities.
The sequence of terms of the sequence followed by their Matula-Goebel trees begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  12: (oo(o))
  16: (oooo)
  17: (((oo)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((o)))
  23: (((o)(o)))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  31: (((((o)))))
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000081, A004111, A007097, A061775, A098859, A130091, A255231, A276625, A316793, A316794, A316795, A316796.

Programs

Mathematica

mgsiQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],And@@Cases[FactorInteger[n],{p_,_}:>mgsiQ[PrimePi[p]]]]];
Select[Range[100],mgsiQ]

PARI

is(n)={my(f=factor(n)); if(#Set(f[,2])<#f~, 0, for(i=1, #f~, if(!is(primepi(f[i,1])), return(0))); 1)}
{ select(is, [1..200]) } \\ Andrew Howroyd, Mar 01 2020

Extensions

Terms a(53) and beyond from Andrew Howroyd, Mar 01 2020