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A256253 Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.

%I A256253 #30 Jun 11 2015 10:26:50
%S A256253 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 A256253 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 A256253 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 A256253 Number of successive odd nonprimes A014076 and number of successive odd primes A065091, interleaved.
%C A256253 See also A256252 and A256262 which contain similar diagrams.
%F A256253 a(n) = A256252(n-1), n >= 3.
%e A256253 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 A256253 1;
%e A256253 3, 5, 7;
%e A256253 9;
%e A256253 11, 13;
%e A256253 15;
%e A256253 17; 19;
%e A256253 21,
%e A256253 23;
%e A256253 25, 27;
%e A256253 39, 31;
%e A256253 ...
%e A256253 a(n) is the length of the n-th row.
%e A256253 .
%e A256253 Illustration of the first 16 regions of the diagram of the symmetric representation of odd nonprimes (A014076) and of odd primes (A065091):
%e A256253 .            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A256253 .           |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |   31
%e A256253 .           |_ _ _ _ _ _ _ _ _ _ _ _ _ _  | |   29
%e A256253 .           | | |_ _ _ _ _ _ _ _ _ _ _  | | |   23
%e A256253 .           | | | |_ _ _ _ _ _ _ _ _  | | | |   19
%e A256253 .           | | | |_ _ _ _ _ _ _ _  | | | | |   17
%e A256253 .           | | | | |_ _ _ _ _ _  | | | | | |   13
%e A256253 .           | | | | |_ _ _ _ _  | | | | | | |   11
%e A256253 .           | | | | | |_ _ _  | | | | | | | |    7
%e A256253 .           | | | | | |_ _  | | | | | | | | |    5
%e A256253 .   A014076 | | | | | |_  | | | | | | | | | |    3
%e A256253 .      1    | | | | | |_|_|_|_| | | | | | | | A065091
%e A256253 .      9    | | | | |_ _ _ _ _|_|_| | | | | |
%e A256253 .     15    | | | |_ _ _ _ _ _ _ _|_|_| | | |
%e A256253 .     21    | | |_ _ _ _ _ _ _ _ _ _ _|_| | |
%e A256253 .     25    | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A256253 .     27    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|
%e A256253 .
%e A256253 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 A256253 .
%e A256253 .                       |_ _ _
%e A256253 .                             |_ _
%e A256253 .                                 |_ _
%e A256253 .                                     |_
%e A256253 .                                       |
%e A256253 .                                       |_ _
%e A256253 .
%e A256253 The sequence begins:    1,3,1,2,1,2,1,1,2,2,...
%e A256253 .
%o A256253 (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 A256253 Cf. A005408, A014076, A047846, A065091, A175632, A251092, A256252, A256134, A256262.
%K A256253 nonn
%O A256253 1,2
%A A256253 _Omar E. Pol_, Mar 30 2015