A128280 a(n) is the least number not occurring earlier such that a(n)+a(n-1) is prime, a(0) = 0.

0, 2, 1, 4, 3, 8, 5, 6, 7, 10, 9, 14, 15, 16, 13, 18, 11, 12, 17, 20, 21, 22, 19, 24, 23, 30, 29, 32, 27, 26, 33, 28, 25, 34, 37, 36, 31, 40, 39, 44, 35, 38, 41, 42, 47, 50, 51, 46, 43, 54, 49, 48, 53, 56, 45, 52, 55, 58, 69, 62, 65, 66, 61, 70, 57, 74, 63, 64, 67, 60, 71, 68, 59

Offset

0,2

Comments

Original definition: start with a(1) = 2. See A055265 for start with a(1) = 1.

The sequence may well be a rearrangement of natural numbers. Interestingly, subsets of first n terms are permutations of 1..n for n = {2, 4, 8, 10, 18, 22, 24, 56, ...}. E.g., first 56 terms: {2, 1, 4, 3, 8, 5, 6, 7, 10, 9, 14, 15, 16, 13, 18, 11, 12, 17, 20, 21, 22, 19, 24, 23, 30, 29, 32, 27, 26, 33, 28, 25, 34, 37, 36, 31, 40, 39, 44, 35, 38, 41, 42, 47, 50, 51, 46, 43, 54, 49, 48, 53, 56, 45, 52, 55} are a permutation of 1..56.

Without altering the definition nor the existing values, one can as well start with a(0) = 0 and get (conjecturally) a permutation of the nonnegative integers. This sequence is in some sense the "arithmetic" analog of the "digital" variant A231433: Here we add subsequent terms, there the digits are concatenated. - M. F. Hasler, Nov 09 2013

The sequence is also a particular case of "among the pairwise sums of any M consecutive terms, N are prime", with M = 2, N = 1. For other M, N see A329333, A329405 ff, A329449 ff and the OEIS Wiki page. - M. F. Hasler, Nov 24 2019

Links

Lars Blomberg, Table of n, a(n) for n = 0..9999

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

Formula

a(2n-1) = A055265(2n-1) + 1, a(2n) = A055265(2n) - 1, for all n >= 1. - M. F. Hasler, Feb 11 2020

Prog

(PARI) {a=0; u=0; for(n=1, 99, u+=1<<a; print1(a", "); for(k=1, 9e9, bittest(u, k)&&next; isprime(a+k)&&(a=k)&&next(2)))}

Crossrefs

Cf. A083236.

Cf. A055265 for the variant starting with a(1) = 1, and A329333, A329405, ..., A329425 and A329449, ..., A329581 for other variants. - M. F. Hasler, Nov 24 2019

Sequence in context: A331740 A074987 A294096 * A182177 A281878 A106625

Adjacent sequences: A128277 A128278 A128279 * A128281 A128282 A128283

Keyword

nonn

Author

Zak Seidov, May 03 2007

Extensions

Initial a(0) = 0 prefixed by M. F. Hasler, Nov 09 2013

Status

approved