Search: id:A100450
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%I A100450
%S A100450 1,7,19,51,99,195,291,483,675,963,1251,1731,2115,2787,3363,4131,4899,
%T A100450 6051,6915,8355,9507,11043,12483,14595,16131,18531,20547,23139,25443,
%U A100450 28803,31107,34947,38019,41859,45315,49923,53379,58851,63171,68547
%N A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and GCD{i,
               j,k} <= 1.
%C A100450 Note that GCD{0,m} = m for any m.
%F A100450 G.f.: (3+Sum(moebius(k)*((1+x^k)/(1-x^k))^3, k=1..infinity))/(1-x). - 
               Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 22 2004. [Sketch of proof: 
               Let b(n) = number of ordered triples (i, j, k) with |i| + |j| + |k| 
               = n and GCD{i, j, k}<= 1. Then a(n) = A100450(n) = partial sums of 
               b(n) and Sum_{d divides n} b(d) = 4*n^2+2 = A005899(n) with g.f. 
               ((1+x)/(1-x))^3.]
%p A100450 f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from 
               -n to n do t2:=gcd(i,j); for k from -n to n do if abs(i) + abs(j) 
               + abs(k) <= n then t3:=gcd(t2,k); if t3 <= 1 then t1:=t1+1; fi; fi; 
               od: od: od: t1; end;
%t A100450 f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] 
               <= n && GCD[i, j, k] <= 1, {i, j, k}, {0, 0, 0}], {i, -n, n}, {j, 
               -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 40}] (from Robert 
               G. Wilson v Dec 14 2004)
%Y A100450 Cf. A100448, A100449, A027430, etc., also A018805.
%Y A100450 Sequence in context: A018728 A027523 A100545 this_sequence A155423 A155347 
               A155385
%Y A100450 Adjacent sequences: A100447 A100448 A100449 this_sequence A100451 A100452 
               A100453
%K A100450 nonn,easy
%O A100450 0,2
%A A100450 N. J. A. Sloane (njas(AT)research.att.com), Nov 21 2004
%E A100450 The keyword "more" refers to the fact that I would also like to get the 
               sequences of the numbers of distinct sums i+j+k (also distinct products 
               i*j*k) over all ordered triples (i,j,k) with |i| + |j| + |k| <= n; 
               also over all ordered triples (i,j,k) with |i| + |j| + |k| <= n and 
               GCD{i,j,k} <= 1.
%E A100450 Also the sequences of the numbers of distinct sums i+j+k (also distinct 
               products i*j*k) over all ordered triples (i,j,k) with i >= 0, j >
               = 0, k >= 0 and i + j + k = n; also over all ordered triples (i,j,
               k) with i >= 0, j >= 0, k >= 0, i + j + k = n and GCD{i,j,k} <= 1.
%E A100450 Also the number of ordered triples (i,j,k) with i >= 0, j >= 0, k >= 
               0, i + j + k = n and GCD{i,j,k} <= 1.

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