A097977 -id:A097977 - cp's OEIS frontend

A191587 Smallest prime p such that p-n is the product of exactly n distinct primes.

3, 17, 73, 1789, 6011, 1616621, 2114977, 111546443, 2156564419, 742073813491, 784009188701, 3093211012526647, 1240404920534023, 3206211009211419509, 171718219950879367781, 993969333554296364281, 6899316550553351234327, 108706781456610360939660763, 29365306848773629524600829, 376147205196163170923414109869

Offset: 1

Michel Lagneau, Jun 07 2011

			a(4) = 1789 because 1789 - 4 = 1785 = 3 * 5 * 7 * 17.

Cf. A097977.

Cf. A000040, A001221.

a191587 n = head [p | p <- dropWhile (<= n) a000040_list,
                      a001221 (p - n) == n]
-- Reinhard Zumkeller, Jun 24 2015

A191587 := proc(n) for i from 1 do p := ithprime(i) ; if A001221(p-n) = n then return p ; end if; end do: end proc: # R. J. Mathar, Jul 01 2011

Table[k := 2; While[Not[Length[FactorInteger[Prime[k] - n]] == n], k++ ]; Prime[k], {n, 1, 8}]

A191587List(m) = local(j=1, p); for(n=1, m, p=prime(j); while(omega(p-n)!=n, p=prime(j++)); print1(p, ", "));
  default(primelimit, 2200000); A191587List(7); \\ Klaus Brockhaus, Jun 21 2011

a(10)-a(16) from Donovan Johnson, Jan 14 2012

a(17)-a(20) from Robert Israel, Mar 27 2020

A345740 a(n) is the least prime p such that Omega(p + n) = n where Omega is A001222, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 5, 131, 43, 15619, 281, 6553, 503, 137771, 3061, 244140613, 8179, 22143361, 401393, 199290359, 491503, 8392333984357, 524269, 3486784381, 2097131, 226640986043, 28311529, 303745269775390601, 113246183, 9885033776809, 469762021, 176518460300597, 805306339, 77737724676061053405079339

Offset: 1

Zak Seidov, Jun 25 2021

nonn

			For n=1, a(1) = 2 as 2+1 = 3 (Omega(2 + 1) = Omega(3) = 1, see A000040(1)).
For n=2, 2+2 = 4 = 2*2 (semiprime, Omega(4) = 2, see A001358(1)).
For n=3, 5+3 = 8 = 2*2*2 (triprime, Omega(8) = 3, see A014612(1)).
For n=4, 131+4 = 135 = 3*3*3*5 (Omega(135) = 4,  see A014613(16)).

David A. Corneth, Table of n, a(n) for n = 1..1049

Cf. A000040, A001222, A001358, A014612, A014613, A053669, A069279, A097977.

Table[k=1;While[PrimeOmega[Prime@k+n]!=n,k++];Prime@k,{n,11}] (* Giorgos Kalogeropoulos, Jun 25 2021 *)

a(n) = my(p=2); while (bigomega(p+n) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jun 26 2021

from sympy import factorint, nextprime, primerange
def Omega(n): return sum(e for f, e in factorint(n).items())
def a(n):
    lb = 2**n
    p = nextprime(max(lb-n, 1) - 1)
    while Omega(p+n) != n: p = nextprime(p)
    return p
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Aug 14 2021

a(n) + n >= A053669(n)^n for n > 2 if a(n) exists. - David A. Corneth, Aug 14 2021

cp's OEIS Frontend

A191587 Smallest prime p such that p-n is the product of exactly n distinct primes.

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