OFFSET
0,1
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
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
FORMULA
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 *)
PROG
(Haskell)
a067970 n = a067970_list !! (n-1)
a067970_list = zipWith (-) (tail a014076_list) a014076_list
-- Reinhard Zumkeller, Sep 30 2011
(Python)
from sympy import primepi, isprime
def A067970(n):
if n == 0: return 8
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, Jul 31 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Feb 04 2002
EXTENSIONS
Edited by Robert G. Wilson v, Feb 08 2002
STATUS
approved