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

1,3

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?

Links

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

Example

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

Prog

(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

Crossrefs

Cf. A000040, A002110, A136437.

Sequence in context: A354576 A049764 A136437 * A140991 A302191 A261712

Adjacent sequences: A137325 A137326 A137327 * A137329 A137330 A137331

Keyword

easy,nonn

Author

Ctibor O. Zizka, Apr 07 2008

Status

approved