cp's OEIS Frontend

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

%I A319272 #12 May 08 2021 08:28:55
%S A319272 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,
%T A319272 45,48,49,50,53,54,56,59,61,63,64,67,68,71,72,75,76,80,81,83,88,89,92,
%U A319272 96,97,98,99,103,107,108,112,121,124,125,127,128,131,135,136,144,147,148
%N A319272 Numbers whose prime multiplicities are distinct and whose prime indices are term of the sequence.
%C A319272 A prime index of n is a number m such that prime(m) divides n.
%C A319272 Also Matula-Goebel numbers of rooted trees in which the multiplicities in the multiset of branches directly under any given node are distinct.
%H A319272 Andrew Howroyd, <a href="/A319272/b319272.txt">Table of n, a(n) for n = 1..1000</a>
%e A319272 36 is not in the sequence because 36 = 2^2 * 3^2 does not have distinct prime multiplicities.
%e A319272 The sequence of terms of the sequence followed by their Matula-Goebel trees begins:
%e A319272    1: o
%e A319272    2: (o)
%e A319272    3: ((o))
%e A319272    4: (oo)
%e A319272    5: (((o)))
%e A319272    7: ((oo))
%e A319272    8: (ooo)
%e A319272    9: ((o)(o))
%e A319272   11: ((((o))))
%e A319272   12: (oo(o))
%e A319272   16: (oooo)
%e A319272   17: (((oo)))
%e A319272   18: (o(o)(o))
%e A319272   19: ((ooo))
%e A319272   20: (oo((o)))
%e A319272   23: (((o)(o)))
%e A319272   24: (ooo(o))
%e A319272   25: (((o))((o)))
%e A319272   27: ((o)(o)(o))
%e A319272   28: (oo(oo))
%e A319272   31: (((((o)))))
%t A319272 mgsiQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],And@@Cases[FactorInteger[n],{p_,_}:>mgsiQ[PrimePi[p]]]]];
%t A319272 Select[Range[100],mgsiQ]
%o A319272 (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)}
%o A319272 { select(is, [1..200]) } \\ _Andrew Howroyd_, Mar 01 2020
%Y A319272 Cf. A000081, A004111, A007097, A061775, A098859, A130091, A255231, A276625, A316793, A316794, A316795, A316796.
%K A319272 nonn
%O A319272 1,2
%A A319272 _Gus Wiseman_, Sep 16 2018
%E A319272 Terms a(53) and beyond from _Andrew Howroyd_, Mar 01 2020