A213207 -id:A213207 - 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.]

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}]

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.

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}]

A225523 Number of distinct products i*j over all pairs (i,j) with |i| + |j| <= n.

Original entry on oeis.org

1, 1, 3, 5, 9, 11, 17, 21, 27, 33, 39, 43, 53, 59, 69, 79, 89, 97, 107, 117, 131, 143, 157, 167, 183, 195, 209, 223, 237, 249, 269, 283, 301, 317, 335, 353, 373, 389, 409, 427, 449, 465, 491, 509, 535, 557, 581, 603, 631, 657, 679

Offset: 0

Robert Price, May 09 2013

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

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

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

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

A225527 Number of distinct products i*j over all pairs (i,j) with i,j>=0 and i+j <= n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 20, 22, 27, 30, 35, 40, 45, 49, 54, 59, 66, 72, 79, 84, 92, 98, 105, 112, 119, 125, 135, 142, 151, 159, 168, 177, 187, 195, 205, 214, 225, 233, 246, 255, 268, 279, 291, 302, 316, 329, 340, 352, 367, 377, 392

Offset: 0

Robert Price, May 09 2013

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

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

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

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

A225529 Number of distinct products i*j over all pairs (i,j) with i,j >= 0 and i+j <= n and gcd(i,j) <= 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 11, 14, 16, 20, 22, 26, 29, 32, 35, 41, 44, 50, 54, 59, 63, 70, 74, 80, 85, 92, 98, 108, 112, 121, 129, 137, 144, 153, 158, 170, 177, 186, 192, 204, 210, 222, 231, 240, 249, 262, 270, 284, 293, 305, 315, 331, 340, 353

Offset: 0

Robert Price, May 09 2013

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.

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.

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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.

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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.

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

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

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PARI

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

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A225523 Number of distinct products i*j over all pairs (i,j) with |i| + |j| <= n.

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A225527 Number of distinct products i*j over all pairs (i,j) with i,j>=0 and i+j <= n.

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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.