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

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, 26, 28, 29, 34, 31, 32, 33, 30, 35, 39, 37, 38, 40, 36, 41, 44, 43, 42, 45, 46, 47, 48, 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, 77, 78

Offset

1,2

Comments

Conjectured to be a permutation of the positive integers.

From M. F. Hasler, Nov 14 2019: (Start)

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.

See A329450, A329452 onward and the wiki page for variants and further considerations about existence, surjectivity, etc. of such sequences. (End)

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..10000

M. F. Hasler, Prime sums from neighboring terms, OEIS wiki, Nov. 23, 2019

Example

a(1) = 1 from minimality.
a(2) = 3 since 2 would produce 3 (a prime) by making 1 + 2.
a(3) = 5 since 2 or 4 would produce a prime (e.g., 3 + 4 = 7).
a(4) = 7 since 2, 4 or 6 would produce a prime (e.g., 5 + 6 = 11).
...
a(8) = 14 as 2, 4, 6, 8, 10 or 12 would produce a prime together with a(7) = 13 or a(6) = 11.
a(9) = 2 as neither 2 + 13 = 15 nor 2 + 14 = 16 is prime.
And so on.

Mathematica

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 *)

Prog

(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).

Crossrefs

Cf. A329333 (3 consecutive terms, exactly 1 prime sum).

Cf. A329406 .. A329410 (exactly 1 prime sum using 4, ..., 10 consecutive terms).

Cf. A329411 .. A329416 (exactly 2 prime sums using 3, ..., 10 consecutive terms).

See also A329450, A329452 onwards for "nonnegative" variants.

Sequence in context: A213924 A082655 A050828 * A194391 A347468 A081534

Adjacent sequences: A329402 A329403 A329404 * A329406 A329407 A329408

Keyword

nonn

Author

Eric Angelini and Jean-Marc Falcoz, Nov 13 2019

Extensions

Edited by N. J. A. Sloane, Nov 15 2019

Status

approved