OFFSET
1,3
COMMENTS
Since this sequence includes 0 no terms are prime. - Charles R Greathouse IV, Jul 25 2013
Lexicographically earliest sequence of distinct natural numbers such that no two terms differ by a prime. - Peter Munn, Jun 19 2017
Congruence analysis from Peter Munn, Jun 30 2017: (Start)
If a(k) is in congruence class q mod p for some prime p, a(k) + p is the only higher number in this class that can be written as prime + a(k). Thus the ways a number m can be written as prime + a(k) for some k are much constrained if m shares membership of one or more such congruence classes with all except a few of the smaller terms in the sequence.
Of the first 100 terms, congruence class 1 mod 2 (odd numbers) contains 95, 1 mod 3 contains 76, and 0 mod 5 contains 53. No other congruence class modulo a prime contains more than 23.
The only even terms up to a(10000) are 0, 10, 34, 100, 310; of which 10, 100 and 310 are congruent to 10 mod 30, therefore to both 1 mod 3 and 0 mod 5. Note an initial sparseness of terms not congruent to either 1 mod 3 or 0 mod 5: this subsequence starts 9, 309, 527, 899, 989, 999. It becomes less sparse: as a proportion of the main sequence it is 0.04, 0.086 and 0.1555 of the first 100, 1000 and 10000 terms respectively.
Conjecture: there are only finitely many even terms.
(End)
LINKS
David W. Wilson, Table of n, a(n) for n = 1..10000.
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved