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Maximal GCD of eight positive integers with sum n.
8

%I #19 Sep 21 2022 11:28:16

%S 1,1,1,1,1,1,1,1,2,1,2,1,2,1,2,1,3,1,2,3,2,1,3,1,4,3,2,1,4,1,2,3,5,1,

%T 3,1,4,5,2,1,6,1,5,3,4,1,6,5,7,3,2,1,6,1,2,7,8,5,6,1,4,3,7,1,9,1,2,5,

%U 4,7,6,1,10,9,2,1,7,5,2,3,11,1,10,7,4,3,2,5,12,1,7,11,10

%N Maximal GCD of eight positive integers with sum n.

%p b:= proc(n, i, t) option remember; `if`(n=0, signum(t),

%p `if`(min(i, t)<1, 1, max(b(n, i-1, t),

%p igcd(b(n-i, min(n-i, i), t-1), i))))

%p end:

%p a:= n-> `if`(n<8, 0, b(n$2, 8)):

%p seq(a(n), n=8..200); # _Alois P. Heinz_, Jul 13 2022

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], If[Min[i, t] < 1, 1, Max[b[n, i - 1, t], GCD[b[n - i, Min[n - i, i], t - 1], i]]]];

%t a[n_] := If[n < 8, 0, b[n, n, 8]];

%t Table[a[n], {n, 8, 100}] (* _Jean-François Alcover_, Sep 21 2022, after _Alois P. Heinz_ *)

%Y Cf. A009694, A162787.

%Y Maximal GCD of k positive integers with sum n for k = 2..10: A032742 (k=2,n>=2), A355249 (k=3), A355319 (k=4), A355366 (k=5), A355368 (k=6), A355402 (k=7), this sequence (k=8), A354599 (k=9), A354601 (k=10).

%K nonn

%O 8,9

%A _Wesley Ivan Hurt_, Jul 08 2022