OFFSET
1,1
COMMENTS
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".
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).
LINKS
Gus Wiseman, Table of n, a(n) for n = 1..10000
EXAMPLE
The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:
0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,
g,bd,be,h,i,cd,af,ag,ce,abd,abe,
j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,
o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,
bce,ak,ch,r,al,am,ci,bj,abh,an,s,
t,ao,abi,ade,acf,u,bk,acg,v,w,df,
bl,abcd,ap,bm,dg,aq,ef,bn,abce,
cj,x,y,eg,ach,bo,z,ar,bde,bcf,*
PROG
(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)
is(n)=isprime(n) && has(n) \\ Charles R Greathouse IV, Aug 03 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 29 2016
STATUS
approved