A137328 a(n) = prime(n) - primorial(k), where k is the greatest number for which primorial(k) <= prime(n).

0, 1, 3, 1, 5, 7, 11, 13, 17, 23, 1, 7, 11, 13, 17, 23, 29, 31, 37, 41, 43, 49, 53, 59, 67, 71, 73, 77, 79, 83, 97, 101, 107, 109, 119, 121, 127, 133, 137, 143, 149, 151, 161, 163, 167, 169, 1, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 61, 67, 71, 73, 83, 97, 101, 103, 107, 121

Offset: 1

Author

Ctibor O. Zizka, Apr 07 2008

Keywords

• easy
• nonn

Comments

Conjecture: Each prime number appears in this sequence at least once.

Is there any general asymptotic formula for the appearance of prime(n) in this sequence?

Examples

			a(6) = prime(6) - primorial(2) = 13 - 6 = 7.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002110, A136437.

Programs

Mathematica

pn=FoldList[Times, 1, Prime[Range[5]]] (* Increase for n>343 *); a[n_]:=Module[{k=1},Until[pn[[k]]>Prime[n],k++];Prime[n]-pn[[k-1]]];Array[a,67] (* James C. McMahon, May 28 2025 *)

PARI

a(n) = {my(p=prime(n), q=1, P=1); until (P > p, q = nextprime(q+1); P *= q;); p - P/q;} \\ Michel Marcus, Mar 14 2022

Python

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    pn, primk, pk, pkplus = 2, 2, 2, 3
    while True:
        while primk * pkplus <= pn:
            primk, pk, pkplus = primk*pkplus, pkplus, nextprime(pkplus)
        yield pn - primk
        pn = nextprime(pn)
print(list(islice(agen(), 67))) # Michael S. Branicky, Mar 14 2022