A291686 - OEIS

%I #25 Feb 02 2021 04:33:35

%S 1,2,3,4,5,6,8,10,11,12,15,16,17,20,22,24,30,31,32,33,34,40,41,44,48,

%T 51,55,59,60,62,64,66,67,68,80,82,83,85,88,93,96,102,109,110,118,120,

%U 123,124,127,128,132,134,136,155,157,160,164,165,166,170,176,177

%N Numbers whose prime indices other than 1 are distinct prime numbers.

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

%H Andrew Howroyd, <a href="/A291686/b291686.txt">Table of n, a(n) for n = 1..1000</a>

%F Sum_{n>=1} 1/a(n) = 2 * Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - _Amiram Eldar_, Feb 02 2021

%e 9 is not in the sequence because the prime indices of 9 = prime(2)*prime(2) are {2,2} which are prime numbers but not distinct.

%e 15 is in the sequence because the prime indices of 15 = prime(2)*prime(3) are {2,3} which are distinct prime numbers.

%e 21 is not in the sequence because the prime indices of 21 = prime(2)*prime(4) are {2,4} which are distinct but not all prime numbers.

%e 24 is in the sequence because the prime indices of 24 = prime(1)*prime(1)*prime(1)*prime(2) are {1,1,1,2} which without the 1s are distinct prime numbers.

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Or[#===1,UnsameQ@@DeleteCases[primeMS[#],1]&&And@@(PrimeQ/@DeleteCases[primeMS[#],1])]&]

%o (PARI) ok(n)={my(t=n>>valuation(n,2)); issquarefree(t) && !#select(p->!isprime(primepi(p)), factor(t)[,1])} \\ _Andrew Howroyd_, Aug 26 2018

%Y Cf. A000009, A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A279790, A281113, A294788, A294788, A301750, A302242, A302243.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 09 2018