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

1, 2, 4, 7, 8, 14, 16, 19, 28, 32, 38, 43, 49, 53, 56, 64, 76, 86, 98, 106, 107, 112, 128, 131, 133, 152, 163, 172, 196, 212, 214, 224, 227, 256, 262, 263, 266, 301, 304, 311, 326, 343, 344, 361, 371, 383, 392, 424, 428, 443, 448, 454, 512, 521, 524, 526, 532

Offset: 1

Author

Gus Wiseman, Dec 05 2018

Keywords

• nonn

Comments

Union of A291636 (Matula-Goebel numbers of series-reduced rooted trees) and A322385.

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.

A multiplicative semigroup: if x and y are in the sequence, then so is x*y. - Robert Israel, Dec 06 2018

Examples

			1 has no prime indices, so the definition is satisfied vacuously. - _Robert Israel_, Dec 07 2018
We have 301 = prime(4) * prime(14). Since 4 and 14 already belong to the sequence, so does 301.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000002, A001462, A007097, A079000, A079254, A214577, A276625, A291636, A304360, A320628, A322385.

Programs

Maple

Res:= 1: S:= {1}:
for n from 2 to 1000 do
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  if F subset S then
    Res:= Res, n;
    if not isprime(n) then S:= S union {n} fi
fi
od:
Res; # Robert Israel, Dec 06 2018

Mathematica

tnpQ[n_]:=With[{m=PrimePi/@First/@If[n==1,{},FactorInteger[n]]},And[!MemberQ[m,_?PrimeQ],And@@tnpQ/@m]]
Select[Range[1000],tnpQ]