A025043 - OEIS

%I #17 Jul 19 2017 20:26:33

%S 0,1,9,10,25,34,35,49,55,85,91,100,115,121,125,133,145,155,169,175,

%T 187,195,205,217,235,247,253,259,265,289,295,301,309,310,319,325,335,

%U 343,355,361,375,385,391,395,403,415,425,445,451,469,475,481,485,493,505,511,515

%N a(n) is not of the form prime + a(k), k < n.

%C Since this sequence includes 0 no terms are prime. - _Charles R Greathouse IV_, Jul 25 2013

%C Lexicographically earliest sequence of distinct natural numbers such that no two terms differ by a prime. - _Peter Munn_, Jun 19 2017

%C Congruence analysis from _Peter Munn_, Jun 30 2017: (Start)

%C 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.

%C 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.

%C 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.

%C Conjecture: there are only finitely many even terms.

%C (End)

%H David W. Wilson, <a href="/A025043/b025043.txt">Table of n, a(n) for n = 1..10000.</a>

%Y Cf. A025044, A072545, A084834, A254337.

%K nonn

%O 1,3

%A _David W. Wilson_