Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #29 May 26 2021 02:33:27
%S 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 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 2,1,2,13,2,10,2,67,2,4,2,7,2,4,2,13,2,28,2
%N Number of successive odd numbers that are not twin primes and number of successive twin primes, interleaved.
%C See also both A256252 and A256253 which contain similar diagrams.
%H Antti Karttunen, <a href="/A256262/b256262.txt">Table of n, a(n) for n = 1..30998</a>
%e 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 1;
%e 3, 5, 7;
%e 9;
%e 11, 13;
%e 15;
%e 17; 19;
%e 21, 23, 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 numbers that are not twin primes (A255763) and of twin primes (A001097).
%e . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | 31
%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ | | 29
%e . | | | | |_ _ _ _ _ _ _ _ _ | | | 19
%e . | | | | |_ _ _ _ _ _ _ _ | | | | 17
%e . | | | | | |_ _ _ _ _ _ | | | | | 13
%e . | | | | | |_ _ _ _ _ | | | | | | 11
%e . | | | | | | |_ _ _ | | | | | | | 7
%e . | | | | | | |_ _ | | | | | | | | 5
%e . A255763 | | | | | | |_ | | | | | | | | | 3
%e . 1 | | | | | | |_|_|_|_| | | | | | | A001097
%e . 9 | | | | | |_ _ _ _ _|_|_| | | | |
%e . 15 | | | | |_ _ _ _ _ _ _ _|_|_| | |
%e . 21 | | | |_ _ _ _ _ _ _ _ _ _ _| | |
%e . 23 | | |_ _ _ _ _ _ _ _ _ _ _ _| | |
%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..8:
%e .
%e . |_ _ _
%e . |_ _
%e . |_ _
%e . |
%e . |
%e . |
%e . |_ _
%e .
%e The sequence begins: 1,3,1,2,1,2,4,2,...
%e .
%o (PARI) istwin(n) = isprime(n) && (isprime(n-2) || isprime(n+2));
%o 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 Cf. A005408, A001097, A256252, A256253, A255763.
%K nonn
%O 1,2
%A _Omar E. Pol_, Mar 31 2015
%E More terms from _Michel Marcus_, May 25 2015