A277098 - OEIS

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 (list; graph; refs; listen; history; text; internal format)

	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`
	FORMULA	`a(n) = A000040(A276625(n)).`
	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	`Subsequence of A302491.` `Cf. A000040, A005117, A079000, A079254, A276625.` `Sequence in context: A178317 A032024 A131741 * A096650 A111107 A186641` `Adjacent sequences: A277095 A277096 A277097 * A277099 A277100 A277101`
	KEYWORD	`nonn`
	AUTHOR	`Gus Wiseman, Sep 29 2016`
	STATUS	`approved`