A164291 -id:A164291 - cp's OEIS frontend

A164290 Sequence of twin prime p where the middle term p+1 has 6 prime factors (here p+2 is the associated twin prime, not listed).

239, 599, 809, 1319, 1487, 2087, 2339, 2969, 3299, 4157, 4271, 4787, 5021, 5099, 5231, 5639, 5849, 6359, 6659, 7307, 7349, 9431, 9767, 10007, 10139, 10331, 10709, 10889, 11069, 11171, 11351, 11549, 11717, 11831, 11969, 12539, 13007, 13337

Offset: 1

Carlos Alves, Aug 12 2009

nonn

This sequence is similar to A060213, A102168, A164289 respectively with 3, 4 and 5 prime factors in the middle number.

These sequences are of the form (p,p+1,p+2) with (p,p+2) twin primes and Omega(p+1)=m with m>=3 (m=1 or m=2 is impossible). Here m=6.

			(239, 240, 241): Omega(240)=Omega(2*2*2*2*3*5)=6 and 239, 241 are twin primes.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A060213, A102168, A164289, A164291.

Omega = If[ # == 1, 0, Apply[Plus, Transpose[FactorInteger[ # ]][[2]]]] &; Wmil = Map[Omega, Range[1, 30000]]; Asequence = Flatten@Position[Partition[Wmil, 3, 1], {1, 6, 1}]
Transpose[Select[Partition[Prime[Range[1600]],2,1],#[[2]]-#[[1]]==2 && PrimeOmega[ #[[1]]+1]==6&]][[1]] (* Harvey P. Dale, May 15 2012 *)

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.

Original entry on oeis.org

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

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

nonn

a(n) <= A164291(n).

			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.

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

Cf. A000720, A001222, A077218, A164291.

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);

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 *)

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

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

cp's OEIS Frontend

A164290 Sequence of twin prime p where the middle term p+1 has 6 prime factors (here p+2 is the associated twin prime, not listed).

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