A277098 Finitary primes. Primes of finitary index.

2, 3, 5, 11, 13, 29, 31, 41, 47, 79, 101, 109, 113, 127, 137, 167, 179, 211, 257, 271, 293, 313, 317, 397, 401, 421, 449, 487, 491, 547, 599, 601, 617, 677, 709, 733, 773, 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

Offset: 1

Author

Gus Wiseman, Sep 29 2016

Keywords

• nonn

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).

Examples

			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,*
		

Links

• Gus Wiseman, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A302491.

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

Programs

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

Formula

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