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A308487 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.

%I A308487 #14 Jun 11 2023 14:20:44
%S A308487 3,11,59,71,239,7,13,103,97,79,127,73,23,31,61,157,373,383,251,89,359,
%T A308487 401,683,701,139,337,283,241,211,631,1471,199,1399,661,113,619,1511,
%U A308487 509,293,953,317,773,1583,863,2423,1831,2251,1933,1381,4057,2803,523,1069,2861,1259,1759,3803,4159,4703
%N A308487 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 A308487 a(n) <= A164291(n).
%H A308487 Robert Israel, <a href="/A308487/b308487.txt">Table of n, a(n) for n = 2..923</a>
%F A308487 A077218(A000720(a(n))) = n.
%e A308487 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 A308487 N:= 100: # to get a(2)..a(N)
%p A308487 V:= Array(2..N): count:= 0:
%p A308487 q:= 3:
%p A308487 while count < N-1 do
%p A308487   p:= q;
%p A308487   q:= nextprime(q);
%p A308487   v:= add(numtheory:-bigomega(t),t=p+1..q-1);
%p A308487   if v > N or V[v] > 0 then next fi;
%p A308487   V[v]:= p; count:= count+1;
%p A308487 od:
%p A308487 convert(V,list);
%t A308487 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 A308487 (PARI) count(start, end) = my(i=0); for(k=start+1, end-1, i+=bigomega(k)); i
%o A308487 a(n) = forprime(p=1, , if(count(p, nextprime(p+1))==n, return(p))) \\ _Felix Fröhlich_, May 31 2019
%Y A308487 Cf. A000720, A001222, A077218, A164291.
%K A308487 nonn
%O A308487 2,1
%A A308487 _J. M. Bergot_ and _Robert Israel_, May 31 2019