A275698 - OEIS

%I #40 Oct 23 2023 01:50:55

%S 2,5,13,133,17293,298995973,89398590973228813,

%T 7992108067998667938125889533702533,

%U 63873791370569400659097694858350356285036046451665934814399129508493

%N a(0) = 2, after that a(n) is 3 plus the least common multiple of previous terms.

%C This sequence could be considered a particular case of a possible two-parameter family of sequences of the form: a(n) = k1 + lcm(a(0),a(1),..,a(n-1)), a(0) = k2, where in this case k1=3 and k2=2. With other choices of k1 and k2 it seems it is possible to generate other sequences such as

%C A129871 with k1 = 1 and k2 = 1,

%C A000058 with k1 = 1 and k2 = 2,

%C A082732 with k1 = 1 and k2 = 3,

%C A000215 with k1 = 2 and k2 = 3,

%C A000324 with k1 = 4 and k2 = 1,

%C A001543 with k1 = 5 and k2 = 1,

%C A001544 with k1 = 6 and k2 = 1,

%C A275664 with k1 = 2 and k2 = 2,

%C A000289 with k1 = 3 and k2 = 1.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2311857">On certain nonlinear recurring sequences</a>, Amer. Math. Monthly 70 (1963), 403-405.

%H S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>

%H Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a> [Local copy]

%F a(n) = 3 + lcm(a(0), a(1), ..., a(n - 1)), a(0) = 2.

%F a(n) = 3 + a(n-1)*(a(n-1)-3), for n > 1. - _Christian Krause_, Oct 17 2023. Proof: Follows from associativity of lcm(...) and the fact that gcd(m,m+3)=1:

%F a(n)-3 = lcm(a(0),a(1),...,a(n-2),a(n-1))

%F        = lcm(lcm(a(0),a(1),...,a(n-2)),a(n-1))

%F        = lcm(a(n-1)-3,a(n-1))

%F        = (a(n-1)-3)*a(n-1).

%t A275698 = {2}; Do[AppendTo[A275698, 3 + LCM@@A275698], {i, 9}]; A275698

%Y Cf. A000058, A000215, A000289, A000324, A001543, A001544, A082732, A275664.

%K nonn

%O 0,1

%A _Andres Cicuttin_, Aug 05 2016