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%I A213212 #22 Oct 27 2023 21:14:20
%S A213212 1,1,1,2,3,5,6,10,12,17,20,26,29,38,44,52,59,72,78,94,104,118,130,149,
%T A213212 160,182,198,221,237,263,278,308,330,361,383,416,438,480,509,546,574,
%U A213212 620,646,699,734,777,816,872,907,969,1012,1071,1117,1190,1233,1307,1361
%N A213212 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.
%C A213212 This sequence is in reply to an extension request made in A100450.
%C A213212 Note that gcd(0,m) = m for any m.
%H A213212 Alois P. Heinz, <a href="/A213212/b213212.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..200 from Robert Price)
%F A213212 a(n) = (A213208(n) + 1)/2.
%p A213212 h:= proc() true end:
%p A213212 b:= proc(n) local c, i, j, p;
%p A213212       c:=0;
%p A213212       for i to iquo(n, 3) do
%p A213212         for j from i to iquo(n-i, 2) do
%p A213212           if igcd(i, j, n-i-j)=1 then p:= i*j*(n-i-j);
%p A213212             if h(p) then h(p):= false; c:=c+1 fi
%p A213212           fi
%p A213212         od
%p A213212       od; c
%p A213212     end:
%p A213212 a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
%p A213212 seq(a(n), n=0..60);  # _Alois P. Heinz_, Mar 02 2013
%t A213212 f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n&& GCD[i, j, k] <= 1, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]
%Y A213212 Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213213.
%K A213212 nonn,easy
%O A213212 0,4
%A A213212 _Robert Price_, Mar 02 2013

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