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A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).

%I A333613 #14 Mar 08 2021 23:36:02
%S A333613 1,2,4,7,15,21,51,78,158,230,568,661,1797,2595,5117,7789,19095,21702,
%T A333613 59892,81801,171329,258028,630942,713093,1887828,2776798,5727675,
%U A333613 8335692,20702970,21420664,62826604,92041835,189376593,281410640,656577018,742729123,2087788417
%N A333613 a(1) = 1; thereafter a(n) = Sum_{k = 1..n} a(k/gcd(n,k)).
%H A333613 Vaclav Kotesovec, <a href="/A333613/b333613.txt">Table of n, a(n) for n = 1..3930</a>
%H A333613 Vaclav Kotesovec, <a href="/A333613/a333613.jpg">Plot of a(n)^(1/n) for n = 1..10000</a>
%F A333613 a(1) = 1; a(n) = Sum_{k = 1..n} a(lcm(n, k)/n).
%F A333613 a(1) = 1; a(n) = Sum_{d|n} Sum_{k = 1..d, gcd(d, k) = 1} a(k).
%p A333613 A333613:= proc(n) option remember;
%p A333613 if n<3 then n;
%p A333613 else add( A333613(lcm(n,j)/n), j = 1..n);
%p A333613 end if; end proc;
%p A333613 seq(A333613(n), n=1..40); # _G. C. Greubel_, Mar 08 2021
%t A333613 a[1] = 1; a[n_] := a[n] = Sum[a[k/GCD[n, k]], {k, n}]; Table[a[n], {n, 37}]
%t A333613 a[1] = 1; a[n_] := a[n] = Sum[Sum[If[GCD[k, d] == 1, a[k], 0], {k, d}], {d, Divisors[n]}]; Table[a[n], {n, 37}]
%o A333613 (Sage)
%o A333613 @CachedFunction
%o A333613 def A333613(n): return 1 if n==1 else sum( A333613(lcm(n, j)/n) for j in (1..n) )
%o A333613 [A333613(n) for n in (1..40)] # _G. C. Greubel_, Mar 08 2021
%Y A333613 Cf. A006874, A045545, A057661, A332791.
%K A333613 nonn
%O A333613 1,2
%A A333613 _Ilya Gutkovskiy_, Mar 28 2020