login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

a(1) = 1, a(n) = a(n-1) / gcd(a(n-1),n) if this gcd is > 1, else a(n) = 3*a(n-1) + n + 1.
2

%I #19 May 14 2024 19:53:12

%S 1,6,2,1,9,3,17,60,20,2,18,3,23,84,28,7,39,13,59,198,66,3,33,11,59,

%T 204,68,17,81,27,113,372,124,62,222,37,1,42,14,7,63,3,53,204,68,34,

%U 150,25,125,5,67,254,816,136,464,58,232,4,72,6,80,40,184,23,135

%N a(1) = 1, a(n) = a(n-1) / gcd(a(n-1),n) if this gcd is > 1, else a(n) = 3*a(n-1) + n + 1.

%C A variant of A133058. The behavior of simple computational models of the form a(1), a(n) = a(n-1) / gcd(a(n-1),n) if this gcd is > 1, else a(n) = X*a(n-1) + Y*n + R, depending on parameters [X, Y, R], shows Wolfram complexity classes for cellular automata.

%H Robert Israel, <a href="/A334943/b334943.txt">Table of n, a(n) for n = 1..10000</a>

%e a(2) = 3*a(1) + 2 + 1 = 6, a(3) = a(2)/3 = 2, a(4) = a(3)/2 = 1, a(5) = 3*a(4) + 5 + 1 = 9, ...

%p N:= 100: # for a(1)..a(N)

%p V:= Vector(N):

%p V[1]:= 1:

%p for n from 2 to N do

%p g:= igcd(V[n-1],n);

%p if g > 1 then V[n]:= V[n-1]/g else V[n]:= 3*V[n-1]+n+1 fi

%p od:

%p convert(V,list); # _Robert Israel_, Jun 22 2020

%t a[1] = 1; a[n_] := a[n] = If[(g = GCD[a[n-1], n]) > 1, a[n-1]/g, 3*a[n-1] + n + 1]; Array[a, 100].

%t nxt[{n_,a_}]:=With[{c=GCD[a,n+1]},{n+1,If[c>1,a/c,3a+n+2]}]; NestList[nxt,{1,1},70][[;;,2]] (* _Harvey P. Dale_, May 14 2024 *)

%o (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(g = gcd(va[n-1], n)); if (g > 1, va[n] = va[n-1]/g, va[n] = 3*va[n-1]+n+1);); va;} \\ _Michel Marcus_, May 17 2020

%o (Magma) a:=[1]; for n in [2..70] do if Gcd(a[n-1], n) eq 1 then Append(~a, 3* a[n-1]+n+1); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; a; // _Marius A. Burtea_, May 17 2020

%Y Cf. A133058.

%K nonn

%O 1,2

%A _Ctibor O. Zizka_, May 17 2020