0,1

In this sequence 8 occurs once, but 2,4,6 may occur several times. No other even number arise. 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 sub-cases. 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 with 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.

Comment from Frank Ellermann: 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). Feb 08, 2002

a(n)=A014076[n+1]-A014076[n]

a = Select[ Range[300], !PrimeQ[ # ] && !EvenQ[ # ] & ]; Table[ a[[n + 1]] - a[[n]], {n, 1, Length[a] - 1} ]

Cf. A014076, A000230.

Sequence in context: A165104 A010527 A102887 this_sequence A003675 A121948 A114141

Adjacent sequences: A067967 A067968 A067969 this_sequence A067971 A067972 A067973

nonn

Labos E. (labos(AT)ana.sote.hu), Feb 04 2002

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 08 2002