%I #20 Aug 03 2023 14:06:56

%S 2,3,5,11,13,29,31,41,47,79,101,109,113,127,137,167,179,211,257,271,

%T 293,313,317,397,401,421,449,487,491,547,599,601,617,677,709,733,773,

%U 811,823,829,907,929,977,991,1033,1063,1109,1187,1231,1259,1297,1361,1429,1483,1489,1559,1609,1621,1741,1759,1831,1871,1889

%N Finitary primes. Primes of finitary index.

%C Definition: prime(n) is a finitary prime iff n is a product of distinct finitary primes, where prime = A000040. This sequence could be described as a "multiplicative Aronson transform" of A005117. (Aronson transforms such as A079000 satisfy "n is in a if and only if a(n) is in b".

%C The composite bijection (finitary primes -> finitary numbers -> finite sets of finitary primes) can be used to construct a natural linear extension SET : N -> F where F is the partially ordered inverse limit of all finite Boolean algebras of finite sets of positive integers. Then a(n) = prime(Prod_p a(p)) where the product is over SET(n).

%H Gus Wiseman, <a href="/A277098/b277098.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000040(A276625(n)).

%e The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:

%e 0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,

%e g,bd,be,h,i,cd,af,ag,ce,abd,abe,

%e j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,

%e o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,

%e bce,ak,ch,r,al,am,ci,bj,abh,an,s,

%e t,ao,abi,ade,acf,u,bk,acg,v,w,df,

%e bl,abcd,ap,bm,dg,aq,ef,bn,abce,

%e cj,x,y,eg,ach,bo,z,ar,bde,bcf,*

%o (PARI) has(p)=if(p<7, 1, my(f=factor(primepi(p))); if(vecmax(f[,2])>1, return(0)); for(i=1,#f~, if(!has(f[i,1]), return(0))); 1)

%o is(n)=isprime(n) && has(n) \\ _Charles R Greathouse IV_, Aug 03 2023

%Y Subsequence of A302491.

%Y Cf. A000040, A005117, A079000, A079254, A276625.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 29 2016