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A256262 Number of successive odd numbers that are not twin primes and number of successive twin primes, interleaved.

%I A256262 #29 May 26 2021 02:33:27
%S A256262 1,3,1,2,1,2,4,2,4,2,7,2,4,2,13,2,1,2,13,2,4,2,13,2,4,2,1,2,13,2,4,2,
%T A256262 13,2,4,2,13,2,16,2,34,2,4,2,13,2,28,2,22,2,13,2,7,2,10,2,7,2,73,2,4,
%U A256262 2,1,2,13,2,10,2,67,2,4,2,7,2,4,2,13,2,28,2
%N A256262 Number of successive odd numbers that are not twin primes and number of successive twin primes, interleaved.
%C A256262 See also both A256252 and A256253 which contain similar diagrams.
%H A256262 Antti Karttunen, <a href="/A256262/b256262.txt">Table of n, a(n) for n = 1..30998</a>
%e A256262 Consider an irregular array in which the odd-indexed rows list successive odd numbers that are not twin primes (A255763) and the even-indexed rows list successive twin primes (A001097), in the sequence of odd numbers (A005408), as shown below:
%e A256262 1;
%e A256262 3, 5, 7;
%e A256262 9;
%e A256262 11, 13;
%e A256262 15;
%e A256262 17; 19;
%e A256262 21, 23, 25, 27;
%e A256262 39, 31;
%e A256262 ...
%e A256262 a(n) is the length of the n-th row.
%e A256262 .
%e A256262 Illustration of the first 16 regions of the diagram of the symmetric representation of odd numbers that are not twin primes (A255763) and of twin primes (A001097).
%e A256262 .            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A256262 .           |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |   31
%e A256262 .           |_ _ _ _ _ _ _ _ _ _ _ _ _ _  | |   29
%e A256262 .           | | | | |_ _ _ _ _ _ _ _ _  | | |   19
%e A256262 .           | | | | |_ _ _ _ _ _ _ _  | | | |   17
%e A256262 .           | | | | | |_ _ _ _ _ _  | | | | |   13
%e A256262 .           | | | | | |_ _ _ _ _  | | | | | |   11
%e A256262 .           | | | | | | |_ _ _  | | | | | | |    7
%e A256262 .           | | | | | | |_ _  | | | | | | | |    5
%e A256262 .   A255763 | | | | | | |_  | | | | | | | | |    3
%e A256262 .      1    | | | | | | |_|_|_|_| | | | | | | A001097
%e A256262 .      9    | | | | | |_ _ _ _ _|_|_| | | | |
%e A256262 .     15    | | | | |_ _ _ _ _ _ _ _|_|_| | |
%e A256262 .     21    | | | |_ _ _ _ _ _ _ _ _ _ _| | |
%e A256262 .     23    | | |_ _ _ _ _ _ _ _ _ _ _ _| | |
%e A256262 .     25    | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e A256262 .     27    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|
%e A256262 .
%e A256262 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..8:
%e A256262 .
%e A256262 .                         |_ _ _
%e A256262 .                               |_ _
%e A256262 .                                   |_ _
%e A256262 .                                       |
%e A256262 .                                       |
%e A256262 .                                       |
%e A256262 .                                       |_ _
%e A256262 .
%e A256262 The sequence begins:      1,3,1,2,1,2,4,2,...
%e A256262 .
%o A256262 (PARI) istwin(n) = isprime(n) && (isprime(n-2) || isprime(n+2));
%o A256262 lista(nn) = {my(nb = 1, istp = 0); forstep (n=3, nn, 2, if (bitxor(istp, ! istwin(n)), nb++, print1(nb, ", "); nb = 1; istp = ! istp););} \\ _Michel Marcus_, May 25 2015
%Y A256262 Cf. A005408, A001097, A256252, A256253, A255763.
%K A256262 nonn
%O A256262 1,2
%A A256262 _Omar E. Pol_, Mar 31 2015
%E A256262 More terms from _Michel Marcus_, May 25 2015