cp's OEIS Frontend

A196274 Half of the gaps A067970 between odd nonprimes A014076.

%I A196274 #27 Oct 24 2024 02:21:12
%S A196274 4,3,3,2,1,3,1,2,3,2,1,2,1,3,1,2,3,1,2,2,1,2,1,1,2,3,3,2,1,1,1,1,1,2,
%T A196274 2,1,3,1,1,1,3,1,2,1,2,2,1,2,1,3,1,1,1,3,3,1,1,1,1,2,1,1,1,1,2,3,2,1,
%U A196274 3,1,1,1,2,1,2,1,2,1,3,1,2,3,1,1,1,2
%N A196274 Half of the gaps A067970 between odd nonprimes A014076.
%C A196274 a(n) < 4 for n > 1; a(A196276(n)) = 1; a(A196277(n)) > 1. - _Reinhard Zumkeller_, Sep 30 2011
%C A196274 Lengths of runs of equal terms in A025549. That sequence begins with: 1,1,1,1,3,3,3,45,45,45,..., that is 4 ones, 3 threes, 3 forty-fives, ... - _Michel Marcus_, Dec 02 2014
%H A196274 Reinhard Zumkeller, <a href="/A196274/b196274.txt">Table of n, a(n) for n = 1..10000</a>
%F A196274 a(n) = (A014076(n+1)-A014076(n))/2 = A067970(n)/2.
%e A196274 The smallest odd numbers which are not prime are 1, 9, 15, 21, 25, 27,... (sequence A014076).
%e A196274 The gaps between these are: 8, 6, 6, 4, 2,... (sequence A067970), which are of course all even by construction, so it makes sense to divide all of them by 2, which yields this sequence: 4, 3, 3, 2, 1, ...
%t A196274 With[{nn=401},Differences[Complement[Range[1,nn,2],Prime[Range[ PrimePi[ nn]]]]]/2] (* _Harvey P. Dale_, May 06 2012 *)
%o A196274 (PARI) L=1;forstep(n=3,299,2,isprime(n)&next;print1((n-L)/2",");L=n)
%o A196274 (Python)
%o A196274 from sympy import primepi, isprime
%o A196274 def A196274(n):
%o A196274     if n == 1: return 4
%o A196274     m, k = n-1, primepi(n) + n - 1 + (n>>1)
%o A196274     while m != k:
%o A196274         m, k = k, primepi(k) + n - 1 + (k>>1)
%o A196274     for d in range(1,4):
%o A196274         if not isprime(m+(d<<1)):
%o A196274             return d # _Chai Wah Wu_, Jul 31 2024
%Y A196274 Cf. A142723 for the decimal value of the associated continued fraction.
%K A196274 nonn
%O A196274 1,1
%A A196274 _M. F. Hasler_, Sep 30 2011
%E A196274 More terms from _Harvey P. Dale_, May 06 2012