1,3

The length (number of decimal digits) of a(n) may be a power of 2 and often simply doubles, when n is increased by 1. But there are many exceptions: n = 11, 12, 13 give lengths 2^8, 3*2^7, 2^9, respectively. A factor of 3 is found in the lengths of a(n) for n = 12, 112..123, 1113..1234, 11123..12345, and so on. A factor of 7 is found for n = 1112, 11112..11122, and so on. 15 is factor of the length of a(11111112).

Using Fermat's little theorem it is easy to prove that for every prime number p there is a smallest n, such that p is a factor of the length of a(n).

Wikipedia "Fermat's little theorem"

R:= (S, m)-> iquo (S+m-1, m): A:= proc (m, n) option remember; `if`(n=1, 1, R (parse (cat (seq (A (m, j), j=1..n-1))), m)) end: a:= n-> A(n, n): seq(a(n), n=1..10);

Cf. A156146, A156147, A000040, A010783.

Sequence in context: A162287 A166168 A126559 this_sequence A164820 A022387 A108559

Adjacent sequences: A159859 A159860 A159861 this_sequence A159863 A159864 A159865

easy,nonn

Eric Angelini (eric.angelini(AT)skynet.be) and Alois P. Heinz (heinz(AT)hs-heilbronn.de), Apr 24 2009