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.

3, 11, 59, 71, 239, 7, 13, 103, 97, 79, 127, 73, 23, 31, 61, 157, 373, 383, 251, 89, 359, 401, 683, 701, 139, 337, 283, 241, 211, 631, 1471, 199, 1399, 661, 113, 619, 1511, 509, 293, 953, 317, 773, 1583, 863, 2423, 1831, 2251, 1933, 1381, 4057, 2803, 523, 1069, 2861, 1259, 1759, 3803, 4159, 4703

Offset: 2

Author

J. M. Bergot and Robert Israel, May 31 2019

Keywords

• nonn

Comments

a(n) <= A164291(n).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..923

Crossrefs

Cf. A000720, A001222, A077218, A164291.

Programs

Maple

N:= 100: # to get a(2)..a(N)
V:= Array(2..N): count:= 0:
q:= 3:
while count < N-1 do
  p:= q;
  q:= nextprime(q);
  v:= add(numtheory:-bigomega(t),t=p+1..q-1);
  if v > N or V[v] > 0 then next fi;
  V[v]:= p; count:= count+1;
od:
convert(V,list);

Mathematica

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 *)

PARI

count(start, end) = my(i=0); for(k=start+1, end-1, i+=bigomega(k)); i
a(n) = forprime(p=1, , if(count(p, nextprime(p+1))==n, return(p))) \\ Felix Fröhlich, May 31 2019

Formula

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