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A067970 First differences of A014076, the odd nonprimes. 7
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 (list; graph; refs; listen; history; text; internal format)
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 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) = 2 * A196274(n); a(A196276(a(n)) = 2; a(A196277(a(n)) > 2. - Reinhard Zumkeller, Sep 30 2011

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

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

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

CROSSREFS

Cf. A014076, A000230.

Sequence in context: A269846 A316136 A102887 * A003675 A254290 A121948

Adjacent sequences:  A067967 A067968 A067969 * A067971 A067972 A067973

KEYWORD

nonn

AUTHOR

Labos Elemer, Feb 04 2002

EXTENSIONS

Edited by Robert G. Wilson v, Feb 08 2002

STATUS

approved

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Last modified July 5 04:30 EDT 2020. Contains 335459 sequences. (Running on oeis4.)