A387864 Numbers r for which there are at least two integers strictly between prime(r) and prime(r+1), all of whose prime factors are less than prime(r+1) - prime(r).

4, 9, 11, 15, 24, 30, 34, 37, 46, 47, 53, 62, 66, 92, 99, 114, 137, 146, 150, 154, 168, 172, 180, 189, 205, 217, 242, 259, 263, 274, 278, 283, 293, 295, 326, 327, 357, 367, 375, 382, 409, 429, 434, 445, 446, 457, 462, 476, 480, 522, 548, 549, 550, 557, 566

Offset

1,1

Comments

Erdős (1976) conjectured that there are infinitely many numbers with this property, but this remains unsolved.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Thomas Bloom, Erdős Problem #932.

P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. See page 29.

Erdős problems database contributors, Erdős problems database. Maintained by Thomas Bloom and Terence Tao.

Formula

a(n) = PrimePi(A387863(n)) = A000720(A383863(n)).

Example

4 is a term because 8 and 9 are two numbers between prime(4) = 7 and prime(5) = 11 whose prime factors are all less than prime(5) - prime(4) = 4.

Prog

(Python)
from itertools import count, islice
from sympy import primerange, nextprime
def A387864_gen(): # generator of terms
    p, q = 3, 5
    for i in count(2):
        plist, c = list(primerange(q-p)), 0
        for n in range(p+1, q):
            for r in plist:
                while True:
                    a, b = divmod(n, r)
                    if not b:
                        n = a
                    else:
                        break
            if n==1:
                c += 1
            if c > 1:
                yield i
                break
        p, q = q, nextprime(q)
A387864_list = list(islice(A387864_gen(), 55)) # Chai Wah Wu, Sep 11 2025

Crossrefs

See A387863 for the corresponding primes.

Sequence in context: A312834 A312835 A396072 * A250176 A312836 A312837

Adjacent sequences: A387861 A387862 A387863 * A387865 A387866 A387867

Keyword

nonn

Author

David Radcliffe, Sep 10 2025

Status

approved