A100380 a(n) = least k such that prime(n) + A002110(k) is prime.

0, 1, 1, 2, 1, 2, 1, 4, 2, 1, 2, 2, 1, 3, 2, 2, 1, 2, 2, 1, 2, 3, 2, 5, 2, 1, 2, 1, 3, 5, 3, 2, 1, 4, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 1, 3, 2, 2, 2, 1, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 3, 2, 3, 2, 2, 3, 2, 3, 4, 3, 3, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 4, 3, 5, 3, 4, 2, 4, 1, 4, 2

Offset: 1

Author

Pierre CAMI, Dec 30 2004

Keywords

• easy
• nonn

Comments

Conjecture: every prime number can be written as +- p(n) -+ p(k)# where p(i)=i-th prime, p(i)#=i-th primorial.

The sequence grows remarkably slowly. The largest number occurring within the first 50000 elements is 90. - Stefan Steinerberger, Apr 10 2006

a(1) = 0 is the minimum value of a(n). It is also unrepeated in this sequence. - Altug Alkan, Dec 02 2015

Examples

			prime(8)=19;
19 + 2 = 21 = 3*7,
19 + 6 = 25 = 5*5, and
19 + 30 = 49 = 7*7, but
19 + 210 = 229, which is prime; 210=prime(4)#, so a(8)=4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (corrected by Ray Chandler, Jan 19 2019)

Crossrefs

Cf. A002110, A265109.

Programs

Maple

primorial:= proc(n) option remember: ithprime(n)*procname(n-1) end proc:
primorial(0):= 1:
f:= proc(n) local k, p;
  p:= ithprime(n);
  for k from 0 do if isprime(p+primorial(k)) then return k fi od:
end proc:
map(f, [$1..100]);# Robert Israel, Aug 27 2015

Mathematica

Table[k := 0;While[Not[PrimeQ[Prime[n]+Product[Prime[i],{i,1,k}]]],k++ ];k,{n,1, 100}] (* Stefan Steinerberger, Apr 10 2006 *)

PARI

primo(n) = prod(i=1, n, prime(i));
a(n) = {k=0; while(!isprime(prime(n)+primo(k)), k++); k;} \\ Michel Marcus, Aug 27 2015

Python

from itertools import count, islice
from sympy import isprime, prime, primorial
def A002110(n): return primorial(n) if n > 0 else 1
def A100380(n):
    pn = prime(n)
    return next(k for k in count(0) if isprime(pn+A002110(k)))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jan 10 2025

Extensions

More terms from Stefan Steinerberger, Apr 10 2006

a(1) = 0 added and name edited by Altug Alkan, Dec 02 2015