A100448 -id:A100448 - cp's OEIS frontend

A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

1, 7, 19, 51, 99, 195, 291, 483, 675, 963, 1251, 1731, 2115, 2787, 3363, 4131, 4899, 6051, 6915, 8355, 9507, 11043, 12483, 14595, 16131, 18531, 20547, 23139, 25443, 28803, 31107, 34947, 38019, 41859, 45315, 49923, 53379, 58851, 63171, 68547

Offset: 0

N. J. A. Sloane, Nov 21 2004

Note that gcd(0,m) = m for any m.
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.
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.
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.
From Robert Price, Mar 05 2013: (Start)
The sequences that address the previous comments are:
Distinct sums i+j+k with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j*k without the GCD qualifier is given by A213207.
Distinct products i*j*k with    the GCD qualifier is given by A213208.
With the restriction i,j,k >= 0 ...
  Distinct sums or products equal to n is trivial and always equals one (A000012).
  Distinct sums <= n results in a(n)=n (A001477).
  Distinct products <= n without the GCD qualifier is given by A213213.
  Distinct products <= n with    the GCD qualifier is given by A213212.
  Ordered triples with sum = n  without the GCD qualifier is A000217(n+1).
  Ordered triples with sum = n  with    the GCD qualifier is A048240.
  Ordered triples with sum <= n without the GCD qualifier is A000292.
  Ordered triples with sum <= n with    the GCD qualifier is A048241. (End)
This sequence (A100450) without the GCD qualifier results in A001845. - Robert Price, Jun 04 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000124, A000292, A018805, A027430, A048240, A048241, A100448, A100449, A213207, A213208, A213212, A213213.

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;

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}] (* Robert G. Wilson v, Dec 14 2004 *)

G.f.: (3 + Sum_{k>=1} (moebius(k)*((1+x^k)/(1-x^k))^3))/(1-x). - Vladeta Jovovic, 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.]

A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 5, 9, 17, 25, 41, 49, 73, 89, 113, 129, 169, 185, 233, 257, 289, 321, 385, 409, 481, 513, 561, 601, 689, 721, 801, 849, 921, 969, 1081, 1113, 1233, 1297, 1377, 1441, 1537, 1585, 1729, 1801, 1897, 1961, 2121, 2169, 2337, 2417, 2513, 2601, 2785, 2849, 3017

Offset: 0

N. J. A. Sloane, Nov 21 2004

nonn

Note that gcd(0,m) = m for any m.
I would also like to get the sequences of the numbers of distinct sums i+j (also distinct products i*j) over all ordered pairs (i,j) with |i| + |j| <= n; also over all ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.
From Robert Price, May 10 2013: (Start)
List of sequences that address these extensions:
Distinct sums i+j with or without the GCD qualifier results in a(n)=2n+1 (A005408).
Distinct products i*j without the GCD qualifier is given by A225523.
Distinct products i*j with    the GCD qualifier is given by A225526.
With the restriction i,j >= 0 ...
Distinct sums or products equal to n is trivial and always equals one (A000012).
Distinct sums <=n with or without the GCD qualifier results in a(n)=n (A001477).
Distinct products <=n without the GCD qualifier is given by A225527.
Distinct products <=n with    the GCD qualifier is given by A225529.
Ordered pairs with the sum = n without the GCD qualifier is a(n)=n+1.
Ordered pairs with the sum = n with    the GCD qualifier is A225530.
Ordered pairs with the sum <=n without the GCD qualifier is A000217(n+1).
Ordered pairs with the sum <=n with    the GCD qualifier is A225531.
(End)
This sequence (A100449) without the GCD qualifier results in A001844. - Robert Price, Jun 04 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A018805, A100448, A100450, A027430, etc.

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 if abs(i) + abs(j) <= n then t2:=gcd(i,j); if t2 <= 1 then t1:=t1+1; fi; fi; od: od: t1; end;
# second Maple program:
b:= proc(n) b(n):= numtheory[phi](n)+`if`(n=0, 0, b(n-1)) end:
a:= n-> 1+4*b(n):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] <= n && GCD[i, j] <= 1, {i, j}, {0, 0}], {i, -n, n}, {j, -n, n}], 1]]]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 14 2004 *)

a(n) = 1+4*sum(k=1, n, eulerphi(k) ); \\ Joerg Arndt, May 10 2013

from functools import lru_cache
@lru_cache(maxsize=None)
def A100449(n):
    if n == 0:
        return 1
    c, j = 0, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*((A100449(k1)-3)//2)
        j, k1 = j2, n//j2
    return 2*(n*(n-1)-c+j)+1 # Chai Wah Wu, Mar 29 2021

a(n) = 1 + 4*Sum(phi(k), k=1..n) = 1 + 4*A002088(n). - Vladeta Jovovic, Nov 25 2004

More terms from Vladeta Jovovic, Nov 25 2004

A213207 Number of distinct products ijk over all triples (i,j,k) with |i| + |j| + |k| <= n.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 13, 19, 25, 35, 43, 55, 65, 79, 91, 111, 127, 149, 167, 193, 217, 249, 273, 311, 339, 383, 419, 463, 501, 551, 591, 643, 693, 751, 799, 869, 925, 995, 1057, 1133, 1199, 1281, 1347, 1439, 1515, 1615, 1697, 1801, 1883, 2001, 2101, 2219, 2313

Offset: 0

Robert Price, Mar 01 2013

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..100 from Robert Price)

Cf. A018805, A027430, A100448, A100449, A100450, A213208, A213212, A213213.

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

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n, {i*j*k}, {0}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 100}]

A213208 Number of distinct products ijk over all triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 11, 19, 23, 33, 39, 51, 57, 75, 87, 103, 117, 143, 155, 187, 207, 235, 259, 297, 319, 363, 395, 441, 473, 525, 555, 615, 659, 721, 765, 831, 875, 959, 1017, 1091, 1147, 1239, 1291, 1397, 1467, 1553, 1631, 1743, 1813, 1937, 2023, 2141, 2233, 2379, 2465

Offset: 0

Robert Price, Mar 01 2013

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

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..100 from Robert Price)

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213212, A213213.

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) +2*b(n)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 01 2013

f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n&& GCD[i, j, k] <= 1, i*j*k, 0], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 100}]

A102309 a(n) = Sum_{d divides n} moebius(d) * binomial(n/d,2).

Original entry on oeis.org

0, 0, 1, 3, 5, 10, 11, 21, 22, 33, 34, 55, 46, 78, 69, 92, 92, 136, 105, 171, 140, 186, 175, 253, 188, 290, 246, 315, 282, 406, 284, 465, 376, 470, 424, 564, 426, 666, 531, 660, 568, 820, 570, 903, 710, 852, 781, 1081, 760, 1155, 890, 1136, 996, 1378, 963, 1420, 1140, 1422, 1246

Offset: 0

Ralf Stephan, Jan 03 2005

nonn

Zero followed by the Moebius transform of A000217. - R. J. Mathar, Jan 19 2009
Apparently, a(n-1) is the number of periodic complex Horadam orbits with period n, for n>2. - Nathaniel Johnston, Oct 04 2013
Also apparently, the first differences of A100448 (checked up to n=2000).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Dorin Andrica and Ovidiu Bagdasar, Recurrent Sequences: Key Results, Applications, and Problems, Springer (2020), p. 159.
Ovidiu Bagdasar, On certain computational and geometric properties of complex Horadam orbits, poster, ANTS 2014.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the characterization of periodic complex Horadam sequences, Fib. Quart. 51 (1) (2013) 28-37.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the Number of Complex Horadam Sequences with a Fixed Period, Fib. Q., 51 (2013), 339-347.
Ovidiu D. Bagdasar and Peter J. Larcombe, On the masked periodicity of Horadam sequences: a generator-based approach, Fib. Q., 55 (2017), 332-339.
Ovidiu Bagdasar and Ioan-Lucian Popa, On the geometry of certain periodic non-homogeneous Horadam sequences, Electronic Notes in Discrete Mathematics 56 (2016) 7-13.

Second column of triangle A020921.

Cf. A002117, A008683, A100448, A326419.

with(numtheory):
a:= n-> add(mobius(d)*binomial(n/d, 2), d=divisors(n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 18 2013

a[n_] := Sum[MoebiusMu[d] Binomial[n/d, 2], {d, Divisors[n]}];
a /@ Range[0, 60] (* Jean-François Alcover, Feb 04 2020 *)

a(n) = sumdiv(n, d, moebius(d) * binomial(n/d,2) ); /* Joerg Arndt, Feb 18 2013 */

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, moebius(k)*x^(2*k)/(1-x^k)^3))) \\ Seiichi Manyama, May 24 2021

from math import comb
from sympy import mobius, divisors
def A102309(n): return sum(mobius(d)*comb(n//d,2) for d in divisors(n,generator=True)) # Chai Wah Wu, May 09 2025

G.f.: Sum_{k>=1} mu(k) * x^(2*k)/(1 - x^k)^3. - Seiichi Manyama, May 24 2021

Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)). - Amiram Eldar, Jun 08 2025

A213212 Number of distinct products ijk over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.

Original entry on oeis.org

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

Robert Price, Mar 02 2013

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

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

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

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

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

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}]

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

A213213 Number of distinct products ijk over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 28, 33, 40, 46, 56, 64, 75, 84, 97, 109, 125, 137, 156, 170, 192, 210, 232, 251, 276, 296, 322, 347, 376, 400, 435, 463, 498, 529, 567, 600, 641, 674, 720, 758, 808, 849, 901, 942, 1001, 1051, 1110, 1157, 1225, 1275

Offset: 0

Robert Price, Mar 02 2013

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from Robert Price)
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212.

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 p:= i*j*(n-i-j);
          if h(p) then h(p):= false; c:=c+1 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

f[n_] := Length[ Union[ Flatten[ Table[ If[ i+j+k <= n, i*j*k, 0], {i, 0, n}, {j, 0, n}, {k, 0, n}], 2]]]; Table[ f[n], {n, 0, 200}]

a(n) = (A213207(n)+1)/2.

A015616 Number of triples (i,j,k) with 1 <= i < j < k <= n and gcd(i,j,k) = 1.

Original entry on oeis.org

0, 0, 1, 4, 10, 19, 34, 52, 79, 109, 154, 196, 262, 325, 409, 493, 613, 712, 865, 997, 1171, 1336, 1567, 1747, 2017, 2251, 2548, 2818, 3196, 3472, 3907, 4267, 4717, 5125, 5665, 6079, 6709, 7222, 7858, 8410, 9190, 9748, 10609, 11299, 12127

Offset: 1

Olivier Gérard

nonn

			For n=6, the a(6) = 19 solutions are the binomial(6,3) = (6*5*4)/(1*2*3) = 20 possible triples minus the triple (2,4,6) with GCD=2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000292, A100448, A027430, A015631.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from 1 to n-2 do for j from i+1 to n-1 do t2:=gcd(i,j); for k from j+1 to n do t3:=gcd(t2,k); if t3 = 1 then t1:=t1+1; fi; od: od: od: t1; end;
# program based on Moebius transform, partial sums of A000741:
with(numtheory):
b:= proc(n) option remember;
      add(mobius(n/d)*(d-2)*(d-1)/2, d=divisors(n))
    end:
a:= proc(n) option remember;
      b(n) +`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 08 2011

a[n_] := (cnt = 0; Do[cnt += Boole[GCD[i, j, k] == 1], {i, 1, n-2}, {j, i+1, n-1}, {k, j+1, n}]; cnt); Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Mar 05 2013 *)

print1(c=0);for(k=1,99,for(j=1,k-1, gcd(j,k)==1 && (c+=j-1) && next; for(i=1,j-1, gcd([i,j,k])>1 || c++)); print1(", "c))

from functools import lru_cache
@lru_cache(maxsize=None)
def A015616(n):
    if n <= 1:
        return 0
    c, j = n*(n-1)*(n-2)//6, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c -= (j2-j)*A015616(k1)
        j, k1 = j2, n//j2
    return c # Chai Wah Wu, Mar 30 2021

a(n) = (A071778(n) - 3*A018805(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004
a(n) = Sum_{i=1..n} A000741(i). - Alois P. Heinz, Feb 08 2011
For n > 1, a(n) = n(n-1)(n-2)/6 - Sum_{j=2..n} a(floor(n/j)) = A000292(n-2) - Sum_{j=2..n} a(floor(n/j)). - Chai Wah Wu, Mar 30 2021

A225531 Number of ordered pairs (i, j) with i, j >= 0, i + j <= n and gcd(i, j) <= 1.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, 66, 74, 82, 98, 104, 122, 130, 142, 152, 174, 182, 202, 214, 232, 244, 272, 280, 310, 326, 346, 362, 386, 398, 434, 452, 476, 492, 532, 544, 586, 606, 630, 652, 698, 714, 756, 776

Offset: 0

Robert Price, May 09 2013

This sequence is in reply to an extension request made in A100449.

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

Robert Price, Table of n, a(n) for n = 0..400

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212, A213213.

f[n_] := Length[Complement[Union[Flatten[Table[If[i + j <= n && GCD[i, j] <= 1, {i, j}], {i, 0, n}, {j, 0, n}], 1]], {Null}]]; Table[f[n], {n, 0, 100}]

alist(N) = my(c=2); vector(N, i, if(1==i, 1, c+=eulerphi(i-1))); \\ Ruud H.G. van Tol, Jul 09 2024

A225530 Number of ordered pairs (i,j) with i,j >= 0, i + j = n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 2, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66

Offset: 0

Robert Price, May 09 2013

This sequence is in reply to an extension request made in A100449.
Note that gcd(0,m) = m for any m.
Apparently a(n) = A000010(n), n >= 2. - R. J. Mathar, May 11 2013

Robert Price, Table of n, a(n) for n = 0..400

Cf. A018805, A027430, A100448, A100449, A100450, A213207, A213208, A213212, A213213.

f[n_]:=Length[Complement[Union[Flatten[Table[If[i+j==n&&GCD[i, j]<=1, {i,j}], {i, 0, n}, {j, 0, n}], 1]], {Null}]]; Table[f[n], {n, 0, 100}]

cp's OEIS Frontend

A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A100449 Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A213207 Number of distinct products i*j*k over all triples (i,j,k) with |i| + |j| + |k| <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A213208 Number of distinct products i*j*k over all triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A102309 a(n) = Sum_{d divides n} moebius(d) * binomial(n/d,2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A213213 Number of distinct products i*j*k over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A213207 Number of distinct products ijk over all triples (i,j,k) with |i| + |j| + |k| <= n.

A213208 Number of distinct products ijk over all triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.

A213212 Number of distinct products ijk over all triples (i,j,k) with i,j,k >= 0 and i+j+k <= n and gcd(i,j,k) <= 1.

A213213 Number of distinct products ijk over all triples (i,j,k) with i,j,k>=0 and i+j+k <= n.