A164510 First differences of A071904 (Odd composite numbers).

6, 6, 4, 2, 6, 2, 4, 6, 4, 2, 4, 2, 6, 2, 4, 6, 2, 4, 4, 2, 4, 2, 2, 4, 6, 6, 4, 2, 2, 2, 2, 2, 4, 4, 2, 6, 2, 2, 2, 6, 2, 4, 2, 4, 4, 2, 4, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 6, 4, 2, 6, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 6, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 6, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 6

Offset: 1

Author

Zak Seidov, Aug 14 2009

Keywords

• nonn
• changed

Comments

Are all terms <=6?

This is A067970 without its first term. [R. J. Mathar, Aug 17 2009]

Yes, all terms are at most 6. For a value of 8, we have to have p, p+2, p+4 all prime, and this is possible only for p=3. As a result, 1 would have to be an odd composite number, which it is not. Therefore all terms are <=6. [J. Lowell, Aug 17 2009]

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Joel E. Cohen and Dexter Senft, Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 494-503. See p. 495.

Crossrefs

Cf. A071904.

Programs

Mathematica

Differences@ Select[Range[1, 360, 2], CompositeQ] (* Michael De Vlieger, Aug 29 2025 *)

Python

from sympy import primepi, isprime
def A164510(n):
    m, k = n, primepi(n+1) + n + (n+1>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    for d in range(2, 7, 2):
        if not isprime(m+d):
            return d # Chai Wah Wu, Aug 02 2024