A196274 Half of the gaps A067970 between odd nonprimes A014076.

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, 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, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 1, 2

Offset: 1

Author

M. F. Hasler, Sep 30 2011

Keywords

• nonn

Comments

a(n) < 4 for n > 1; a(A196276(n)) = 1; a(A196277(n)) > 1. - Reinhard Zumkeller, Sep 30 2011

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

Examples

			The smallest odd numbers which are not prime are 1, 9, 15, 21, 25, 27,... (sequence A014076).
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, ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A142723 for the decimal value of the associated continued fraction.

Programs

Mathematica

With[{nn=401},Differences[Complement[Range[1,nn,2],Prime[Range[ PrimePi[ nn]]]]]/2] (* Harvey P. Dale, May 06 2012 *)

PARI

L=1;forstep(n=3,299,2,isprime(n)&next;print1((n-L)/2",");L=n)

Python

from sympy import primepi, isprime
def A196274(n):
    if n == 1: return 4
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    for d in range(1,4):
        if not isprime(m+(d<<1)):
            return d # Chai Wah Wu, Jul 31 2024

Formula

a(n) = (A014076(n+1)-A014076(n))/2 = A067970(n)/2.

Extensions

More terms from Harvey P. Dale, May 06 2012