A067970 First differences of A014076, the odd nonprimes.

8, 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

Offset: 1

Author

Labos Elemer, Feb 04 2002

Keywords

• nonn

Comments

In this sequence 8 occurs once, but 2,4,6 may occur several times. No other even number arises. Therefore sequence consists of {8,6,4,2}.

Proof: If x is an odd nonprime, then x+2=next-odd-number is either nonprime[Case1] or it is a prime [Case 2]. In Case 1 the difference is 2. E.g., x=25, x+2=27, the actual difference is d=2.

In Case 2 x+2=p=prime. Distinguish two further subcases. In Case 2a: x+2=p=prime and p+2=x+4=q is also a prime. Then x+2+2+2=x+6 will not be prime because in first difference sequence of prime no d=2 occurs twice except for p+2=3+2=5,5+2=7, i.e., when p is divisible by 3; for 6k+1 and 6k+5 primes it is impossible. Consequently x+6 is not a prime and so the difference between two consecutive odd nonprimes is 6. Example: x=39, x+2=41=smaller twin prime and next odd nonprime x+6=45, d=6

In Case 2b: x+2=p=prime, but x+2+2=x+4 is not a prime, i.e., x+2=p is not a smaller one of a twin-prime pair. Thus x+4 is the next odd nonprime. Thus the difference=4. E.g., x=77, x+2=79, so the next odd nonprime is x+4=81, d=4. No more cases. QED.

Interestingly this sequence picks out the twin primes.

That the first term is special is a reflection of the simple fact that there are no 3 consecutive odd primes except from 3, 5, 7 corresponding to A067970(1) = 8 = 9-1 = (7+2)-(3-2). - Frank Ellermann, Feb 08 2002

There are arbitrarily long runs of 2's, but not of 4's or 6's. - Zak Seidov, Oct 01 2011

Links

• Jason Yuen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014076, A000230.

Cf. A196274, A196276, A196277.

Programs

Haskell

a067970 n = a067970_list !! (n-1)
a067970_list = zipWith (-) (tail a014076_list) a014076_list
-- Reinhard Zumkeller, Sep 30 2011

Mathematica

a = Select[ Range[300], !PrimeQ[ # ] && !EvenQ[ # ] & ]; Table[ a[[n + 1]] - a[[n]], {n, 1, Length[a] - 1} ]
With[{nn=401},Differences[Complement[Range[1,nn,2],Prime[Range[PrimePi [nn]]]]]]  (* Harvey P. Dale, Feb 05 2012 *)

Python

from sympy import primepi, isprime
def A067970(n):
    if n == 1: return 8
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    for d in range(2,7,2):
        if not isprime(m+d):
            return d # Chai Wah Wu, Jul 31 2024

Formula

a(n) = A014076(n+1) - A014076(n).

a(n) = 2 * A196274(n); a(A196276(n)) = 2; a(A196277(n)) > 2. - Reinhard Zumkeller, Sep 30 2011

Extensions

Edited by Robert G. Wilson v, Feb 08 2002

Offset changed to 1 by Jason Yuen, Jan 08 2025