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

1,1

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

Links

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

Formula

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

Example

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, ...

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A177038 A019975 A327869 * A073871 A120927 A241180

Adjacent sequences: A196271 A196272 A196273 * A196275 A196276 A196277

Keyword

nonn

Author

M. F. Hasler, Sep 30 2011

Extensions

More terms from Harvey P. Dale, May 06 2012

Status

approved