cp's OEIS Frontend

A322386 Numbers whose prime indices are not prime and already belong to the sequence.

%I A322386 #15 Dec 07 2018 12:28:28
%S A322386 1,2,4,7,8,14,16,19,28,32,38,43,49,53,56,64,76,86,98,106,107,112,128,
%T A322386 131,133,152,163,172,196,212,214,224,227,256,262,263,266,301,304,311,
%U A322386 326,343,344,361,371,383,392,424,428,443,448,454,512,521,524,526,532
%N A322386 Numbers whose prime indices are not prime and already belong to the sequence.
%C A322386 Union of A291636 (Matula-Goebel numbers of series-reduced rooted trees) and A322385.
%C A322386 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.
%C A322386 A multiplicative semigroup: if x and y are in the sequence, then so is x*y. - _Robert Israel_, Dec 06 2018
%H A322386 Robert Israel, <a href="/A322386/b322386.txt">Table of n, a(n) for n = 1..10000</a>
%e A322386 1 has no prime indices, so the definition is satisfied vacuously. - _Robert Israel_, Dec 07 2018
%e A322386 We have 301 = prime(4) * prime(14). Since 4 and 14 already belong to the sequence, so does 301.
%p A322386 Res:= 1: S:= {1}:
%p A322386 for n from 2 to 1000 do
%p A322386   F:= map(numtheory:-pi, numtheory:-factorset(n));
%p A322386   if F subset S then
%p A322386     Res:= Res, n;
%p A322386     if not isprime(n) then S:= S union {n} fi
%p A322386 fi
%p A322386 od:
%p A322386 Res; # _Robert Israel_, Dec 06 2018
%t A322386 tnpQ[n_]:=With[{m=PrimePi/@First/@If[n==1,{},FactorInteger[n]]},And[!MemberQ[m,_?PrimeQ],And@@tnpQ/@m]]
%t A322386 Select[Range[1000],tnpQ]
%Y A322386 Cf. A000002, A001462, A007097, A079000, A079254, A214577, A276625, A291636, A304360, A320628, A322385.
%K A322386 nonn
%O A322386 1,2
%A A322386 _Gus Wiseman_, Dec 05 2018