A308487 - OEIS

%I #14 Jun 11 2023 14:20:44

%S 3,11,59,71,239,7,13,103,97,79,127,73,23,31,61,157,373,383,251,89,359,

%T 401,683,701,139,337,283,241,211,631,1471,199,1399,661,113,619,1511,

%U 509,293,953,317,773,1583,863,2423,1831,2251,1933,1381,4057,2803,523,1069,2861,1259,1759,3803,4159,4703

%N a(n) is the least prime p such that the total number of prime factors, with multiplicity, of the numbers between p and the next prime is n.

%C a(n) <= A164291(n).

%H Robert Israel, <a href="/A308487/b308487.txt">Table of n, a(n) for n = 2..923</a>

%F A077218(A000720(a(n))) = n.

%e a(8) = 13 because between 13 and the next prime, 17, are 14 with 2 prime factors, 15 with 2, 16 with 4 (counted with multiplicity), for a total of 2+2+4=8, and this is the first prime for which the total of 8 occurs.

%p N:= 100: # to get a(2)..a(N)

%p V:= Array(2..N): count:= 0:

%p q:= 3:

%p while count < N-1 do

%p   p:= q;

%p   q:= nextprime(q);

%p   v:= add(numtheory:-bigomega(t),t=p+1..q-1);

%p   if v > N or V[v] > 0 then next fi;

%p   V[v]:= p; count:= count+1;

%p od:

%p convert(V,list);

%t Module[{nn=60,pfm},pfm=Table[{p,Total[PrimeOmega[Range[Prime[p]+1,Prime[ p+1]-1]]]},{p,2,1000}];Prime[#]&/@Table[SelectFirst[pfm,#[[2]]==n&],{n,2,nn}]][[All,1]] (* _Harvey P. Dale_, Aug 25 2022 *)

%o (PARI) count(start, end) = my(i=0); for(k=start+1, end-1, i+=bigomega(k)); i

%o a(n) = forprime(p=1, , if(count(p, nextprime(p+1))==n, return(p))) \\ _Felix Fröhlich_, May 31 2019

%Y Cf. A000720, A001222, A077218, A164291.

%K nonn

%O 2,1

%A _J. M. Bergot_ and _Robert Israel_, May 31 2019