1,1

For n >= 1, a(n) >= A056830(n), the least n-digit positive integer with no identical adjacent digits (also the least positive integer whose digits occur in n runs). Conjecture: For all n, a(n) <> 0.

If the conjecture is true, then this sequence and the following two sequences are equivalent: i) Smallest prime with exactly n runs of digits, and ii) Smallest prime with at least n runs of digits. For each n <= 625, a(n) is an n-digit prime (provided that each probable prime shown in the link is indeed a prime -- or at least one of very many (slightly) larger probable prime candidates is prime).

As each a(n) shown is very near A056830(n), I believe it is extremely unlikely that a randomly-given n would yield a 0 term (but I don't have a proof for arbitrary n).

Rick L. Shepherd, Table of n, a(n) for n = 1..625

a(4) = 1013 because 1013 is the smallest 4-digit prime having no identical adjacent digits; the only smaller 4-digit prime, 1009, is disqualified by the "00", identical adjacent digits (of run length 2). Also each digit, 1, 0, 1, 3, occurs in a run of identical digits of length 1 for a total of 4 runs with 1013 being the smallest prime of any length with 4 runs of digits.

Cf. A003617, A007809, A056830.

Adjacent sequences: A141113 A141114 A141115 this_sequence A141117 A141118 A141119

Sequence in context: A064325 A123619 A030519 this_sequence A077246 A107000 A046891

base,nonn

Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 05 2008