cp's OEIS Frontend

A277098 Finitary primes. Primes of finitary index.

%I A277098 #20 Aug 03 2023 14:06:56
%S A277098 2,3,5,11,13,29,31,41,47,79,101,109,113,127,137,167,179,211,257,271,
%T A277098 293,313,317,397,401,421,449,487,491,547,599,601,617,677,709,733,773,
%U A277098 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 A277098 Finitary primes. Primes of finitary index.
%C A277098 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 A277098 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 A277098 Gus Wiseman, <a href="/A277098/b277098.txt">Table of n, a(n) for n = 1..10000</a>
%F A277098 a(n) = A000040(A276625(n)).
%e A277098 The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:
%e A277098 0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,
%e A277098 g,bd,be,h,i,cd,af,ag,ce,abd,abe,
%e A277098 j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,
%e A277098 o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,
%e A277098 bce,ak,ch,r,al,am,ci,bj,abh,an,s,
%e A277098 t,ao,abi,ade,acf,u,bk,acg,v,w,df,
%e A277098 bl,abcd,ap,bm,dg,aq,ef,bn,abce,
%e A277098 cj,x,y,eg,ach,bo,z,ar,bde,bcf,*
%o A277098 (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 A277098 is(n)=isprime(n) && has(n) \\ _Charles R Greathouse IV_, Aug 03 2023
%Y A277098 Subsequence of A302491.
%Y A277098 Cf. A000040, A005117, A079000, A079254, A276625.
%K A277098 nonn
%O A277098 1,1
%A A277098 _Gus Wiseman_, Sep 29 2016