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.

1, 1, 1, 2, 3, 5, 6, 10, 12, 17, 20, 26, 29, 38, 44, 52, 59, 72, 78, 94, 104, 118, 130, 149, 160, 182, 198, 221, 237, 263, 278, 308, 330, 361, 383, 416, 438, 480, 509, 546, 574, 620, 646, 699, 734, 777, 816, 872, 907, 969, 1012, 1071, 1117, 1190, 1233, 1307, 1361

Offset: 0

Author

Robert Price, Mar 02 2013

Keywords

• nonn
• easy

Comments

This sequence is in reply to an extension request made in A100450.

Note that gcd(0,m) = m for any m.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)

Crossrefs

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213213.

Programs

Maple

h:= proc() true end:
b:= proc(n) local c, i, j, p;
      c:=0;
      for i to iquo(n, 3) do
        for j from i to iquo(n-i, 2) do
          if igcd(i, j, n-i-j)=1 then p:= i*j*(n-i-j);
            if h(p) then h(p):= false; c:=c+1 fi
          fi
        od
      od; c
    end:
a:= proc(n) a(n):= `if`(n=0, 1, a(n-1) +b(n)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 02 2013

Mathematica

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}]

Formula

a(n) = (A213208(n) + 1)/2.