Search: id:A067970
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%I A067970
%S A067970 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,
%T A067970 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,
%U A067970 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
%N A067970 First differences of A014076, the odd nonprimes.
%C A067970 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}.
%C A067970 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.
%C A067970 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
%C A067970 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.
%C A067970 Interestingly this sequence picks out the twin primes.
%C A067970 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
%F A067970 a(n)=A014076[n+1]-A014076[n]
%t A067970 a = Select[ Range[300], !PrimeQ[ # ] && !EvenQ[ # ] & ]; Table[ a[[n 
               + 1]] - a[[n]], {n, 1, Length[a] - 1} ]
%Y A067970 Cf. A014076, A000230.
%Y A067970 Sequence in context: A165104 A010527 A102887 this_sequence A003675 A121948 
               A114141
%Y A067970 Adjacent sequences: A067967 A067968 A067969 this_sequence A067971 A067972 
               A067973
%K A067970 nonn
%O A067970 0,1
%A A067970 Labos E. (labos(AT)ana.sote.hu), Feb 04 2002
%E A067970 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 08 2002

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