A025043 a(n) is not of the form prime + a(k), k < n.

0, 1, 9, 10, 25, 34, 35, 49, 55, 85, 91, 100, 115, 121, 125, 133, 145, 155, 169, 175, 187, 195, 205, 217, 235, 247, 253, 259, 265, 289, 295, 301, 309, 310, 319, 325, 335, 343, 355, 361, 375, 385, 391, 395, 403, 415, 425, 445, 451, 469, 475, 481, 485, 493, 505, 511, 515

Offset: 1

Author

David W. Wilson

Keywords

• nonn

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

Cf. A025044, A072545, A084834, A254337.