%I #30 Jun 11 2015 10:26:50

%S 1,3,1,2,1,2,1,1,2,2,2,1,1,2,1,1,2,1,2,2,2,1,1,2,2,1,1,1,2,1,3,1,1,2,

%T 1,2,1,1,6,1,1,1,2,2,4,2,2,1,2,1,1,1,2,1,2,2,4,2,1,2,5,1,5,1,1,2,1,1,

%U 2,2,4,1,2,1,2,1,2,2,2,1,1,2,4,1,6,1,1,2,1,1,6,1,2,1,4,2,1,1,2,1,3,1,2,1,2

%N Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.

%C See also A256252 and A256262 which contain similar diagrams.

%F a(n) = A256252(n-1), n >= 3.

%e Consider an irregular array in which the odd-indexed rows list successive odd nonprimes (A014076) and the even-indexed rows list successive odd primes (A065091), in the sequence of odd numbers (A005408), as shown below:

%e 1;

%e 3, 5, 7;

%e 9;

%e 11, 13;

%e 15;

%e 17; 19;

%e 21,

%e 23;

%e 25, 27;

%e 39, 31;

%e ...

%e a(n) is the length of the n-th row.

%e .

%e Illustration of the first 16 regions of the diagram of the symmetric representation of odd nonprimes (A014076) and of odd primes (A065091):

%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | 31

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ | | 29

%e . | | |_ _ _ _ _ _ _ _ _ _ _ | | | 23

%e . | | | |_ _ _ _ _ _ _ _ _ | | | | 19

%e . | | | |_ _ _ _ _ _ _ _ | | | | | 17

%e . | | | | |_ _ _ _ _ _ | | | | | | 13

%e . | | | | |_ _ _ _ _ | | | | | | | 11

%e . | | | | | |_ _ _ | | | | | | | | 7

%e . | | | | | |_ _ | | | | | | | | | 5

%e . A014076 | | | | | |_ | | | | | | | | | | 3

%e . 1 | | | | | |_|_|_|_| | | | | | | | A065091

%e . 9 | | | | |_ _ _ _ _|_|_| | | | | |

%e . 15 | | | |_ _ _ _ _ _ _ _|_|_| | | |

%e . 21 | | |_ _ _ _ _ _ _ _ _ _ _|_| | |

%e . 25 | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |

%e . 27 |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|

%e .

%e a(n) is also the length of the n-th boundary segment in the zig-zag path of the above diagram, between the two types of numbers, as shown below for n = 1..10:

%e .

%e . |_ _ _

%e . |_ _

%e . |_ _

%e . |_

%e . |

%e . |_ _

%e .

%e The sequence begins: 1,3,1,2,1,2,1,1,2,2,...

%e .

%o (PARI) lista(nn) = {my(nb = 1, isp = 0); forstep (n=3, nn, 2, if (bitxor(isp, ! isprime(n)), nb++, print1(nb, ", "); nb = 1; isp = ! isp););} \\ _Michel Marcus_, May 25 2015

%Y Cf. A005408, A014076, A047846, A065091, A175632, A251092, A256252, A256134, A256262.

%K nonn

%O 1,2

%A _Omar E. Pol_, Mar 30 2015