cp's OEIS Frontend

A329405 Among the pairwise sums of any three consecutive terms there is no prime: lexicographically earliest such sequence of distinct positive integers.

%I A329405 #55 May 26 2021 03:08:12
%S A329405 1,3,5,7,9,11,13,14,2,4,6,8,10,12,15,18,17,16,19,20,25,24,21,27,23,22,
%T A329405 26,28,29,34,31,32,33,30,35,39,37,38,40,36,41,44,43,42,45,46,47,48,51,
%U A329405 54,57,58,53,52,59,56,49,50,55,60,61,62,63,66,67,68,65,64,69,71,72,70,73,74,79,76,77,78
%N A329405 Among the pairwise sums of any three consecutive terms there is no prime: lexicographically earliest such sequence of distinct positive integers.
%C A329405 Conjectured to be a permutation of the positive integers.
%C A329405 From _M. F. Hasler_, Nov 14 2019: (Start)
%C A329405 Equivalently: For any n, neither a(n) + a(n+1) nor a(n) + a(n+2) is prime. Or: For any n and 0 <= i < j <= 2, a(n+i) + a(n+j) is never prime.
%C A329405 See A329450, A329452 onward and the wiki page for variants and further considerations about existence, surjectivity, etc. of such sequences. (End)
%H A329405 Jean-Marc Falcoz, <a href="/A329405/b329405.txt">Table of n, a(n) for n = 1..10000</a>
%H A329405 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 A329405 a(1) = 1 from minimality.
%e A329405 a(2) = 3 since 2 would produce 3 (a prime) by making 1 + 2.
%e A329405 a(3) = 5 since 2 or 4 would produce a prime (e.g., 3 + 4 = 7).
%e A329405 a(4) = 7 since 2, 4 or 6 would produce a prime (e.g., 5 + 6 = 11).
%e A329405 ...
%e A329405 a(8) = 14 as 2, 4, 6, 8, 10 or 12 would produce a prime together with a(7) = 13 or a(6) = 11.
%e A329405 a(9) = 2 as neither 2 + 13 = 15 nor 2 + 14 = 16 is prime.
%e A329405 And so on.
%t A329405 a[1]=1;a[2]=3;a[n_]:=a[n]=(k=1;While[Or@@PrimeQ[Plus@@@Subsets[{a[n-1],a[n-2],++k},{2}]]||MemberQ[Array[a,n-1],k]];k);Array[a,100] (* _Giorgos Kalogeropoulos_, May 09 2021 *)
%o A329405 (PARI) A329405(n, show=1, o=1, 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=0: don't print the list; o=0: start with a(0) = 0, i.e., compute A329450. See the wiki page for more general code returning a vector: S(n,0,3,1) = a(1..n).
%Y A329405 Cf. A329333 (3 consecutive terms, exactly 1 prime sum).
%Y A329405 Cf. A329406 .. A329410 (exactly 1 prime sum using 4, ..., 10 consecutive terms).
%Y A329405 Cf. A329411 .. A329416 (exactly 2 prime sums using 3, ..., 10 consecutive terms).
%Y A329405 See also A329450, A329452 onwards for "nonnegative" variants.
%K A329405 nonn
%O A329405 1,2
%A A329405 _Eric Angelini_ and _Jean-Marc Falcoz_, Nov 13 2019
%E A329405 Edited by _N. J. A. Sloane_, Nov 15 2019