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

2,1

Comments

a(n) <= A164291(n).

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000720, A001222, A077218, A164291.

Sequence in context: A071698 A089188 A086827 * A164291 A137690 A107007

Adjacent sequences: A308484 A308485 A308486 * A308488 A308489 A308490

Keyword

nonn

Author

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

Status

approved