This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329450 #40 Dec 21 2024 11:57:50
%S A329450 0,1,8,7,2,13,12,3,6,9,15,5,10,4,11,14,16,18,17,21,19,23,25,26,20,22,
%T A329450 24,27,28,29,34,31,32,33,30,35,39,37,38,40,36,41,44,43,42,45,46,47,48,
%U A329450 51,54,57,58,53,52,59,56,49,50,55,60,61,62,63,66,67,68,65,64,69,71,72,70,73,74,79,76
%N A329450 Lexicographically earliest sequence of distinct nonnegative integers such that neither a(n) + a(n+1) nor a(n) + a(n+2) is prime for any n.
%C A329450 Equivalently: For any three consecutive terms, there is no prime among any of the pairwise sums. Or: For any n and 0 <= i < j <= 2, a(n+i) + a(n+j) is never prime.
%C A329450 For any n, a term a(n) which meets the requirements always exists: For any a(n-2), a(n-1), at least one in five consecutive values of k is such that one among {a(n-2) + k, a(n-1) + k} is divisible by 2 and the other one by 3.
%C A329450 Conjectured to be a permutation of the nonnegative integers. The restriction to positive indices is then a permutation of the positive integers with the same property, but not the lexicographically earliest given in A329405.
%C A329450 See the wiki page for additional considerations and other variants. - _M. F. Hasler_, Nov 24 2019
%H A329450 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2019/11/prime-sums-from-neighbouring-terms.html">Prime sums from neighbouring terms</a>, personal blog "Cinquante signes" (and post to the SeqFan list), Nov. 11, 2019.
%H A329450 Eric Angelini, <a href="/A329333/a329333.htm">Prime sums from neighbouring terms</a> [Cached copy of html file, with permission]
%H A329450 Eric Angelini, <a href="/A329333/a329333.pdf">Prime sums from neighbouring terms</a> [Cached copy of pdf file, with permission]
%H A329450 M. F. Hasler, <a href="/wiki/User:M._F._Hasler/Prime_sums_from_neighboring_terms">Prime sums from neighboring terms</a>, OEIS wiki, Nov. 23, 2019
%e A329450 After the smallest possible initial terms, a(0) = 0, a(1) = 1, the next term must be neither a prime nor a prime - 1. The smallest possibility is a(2) = 8.
%e A329450 The next term must not be a prime - 1 nor a prime - 8, which excludes 2, 4, 6 on one hand, and 3 and 5 on the other hand. The smallest possibility is a(3) = 7.
%t A329450 Nest[Block[{k = 2}, While[Nand[FreeQ[#, k], ! PrimeQ[#[[-1]] + k], ! PrimeQ[#[[-2]] + k]], k++]; Append[#, k]] &, {0, 1}, 89] (* _Michael De Vlieger_, Nov 15 2019 *)
%o A329450 (PARI) A329450(n,show=0,o=0,p=o,U=[])={for(n=o,n-1, show&&print1(p","); U=setunion(U,[p]); while(#U>1&&U[1]==U[2]-1, U=U[^1]); for(k=U[1]+1,oo, setsearch(U,k) || isprime(o+k) || isprime(p+k) || [o=p, p=k, break]));p} \\ Optional args: show=1: print a(o..n-1); o=1: start with a(1) = 1 (A329405). See the wiki page for more general code returning a vector: S(n,0,3) = A329450(0..n-1).
%Y A329450 Cf. A329333 (always one odd prime among a(n+i)+a(n+j), 0 <= i < j <= 2).
%Y A329450 Cf. A329405 (analog for positive integers).
%K A329450 nonn
%O A329450 0,3
%A A329450 _M. F. Hasler_, based on an idea of _Eric Angelini_, Nov 13 2019
%E A329450 Edited by _N. J. A. Sloane_, Nov 14 2019
%E A329450 New definition corrected by _M. F. Hasler_, Nov 15 2019