A027430 -id:A027430 - 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.]

A100448 Number of triples (i,j,k) with 1 <= i <= j < k <= n and gcd{i,j,k} = 1.

Original entry on oeis.org

0, 1, 4, 9, 19, 30, 51, 73, 106, 140, 195, 241, 319, 388, 480, 572, 708, 813, 984, 1124, 1310, 1485, 1738, 1926, 2216, 2462, 2777, 3059, 3465, 3749, 4214, 4590, 5060, 5484, 6048, 6474, 7140, 7671, 8331, 8899, 9719, 10289, 11192, 11902, 12754, 13535, 14616

Offset: 1

N. J. A. Sloane, Nov 21 2004

Probably the partial sums of A102309. - Ralf Stephan, Jan 03 2005

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A015616, A015631, A027430, A071778, A018805.

f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from 1 to n do for j from i to n 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;

f[n_] := Length[ Union[ Flatten[ Table[ If[ GCD[i, j, k] == 1, {i, j, k}], {i, n}, {j, i, n}, {k, j + 1, n}], 2]]]; Table[ If[n > 3, f[n] - 1, f[n]], {n, 47}] (* Robert G. Wilson v, Dec 14 2004 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A100448(n):
    if n == 0:
        return 0
    c, j = 2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(6*A100448(k1)+1)
        j, k1 = j2, n//j2
    return (n*(n**2-1)-c+j)//6 # Chai Wah Wu, Mar 29 2021

a(n) = (A071778(n)-1)/6. - Vladeta Jovovic, Nov 30 2004

a(n) = (1/6)*(-1 + Sum_{k=1..n} moebius(k)*floor(n/k)^3). - Ralf Stephan, Jan 03 2005

More terms from Robert G. Wilson v, Dec 14 2004

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A027425 Number of distinct products ijk with 1 <= i,j,k <= n.

Original entry on oeis.org

1, 4, 10, 16, 30, 40, 65, 80, 100, 120, 173, 194, 266, 301, 343, 378, 492, 536, 678, 732, 804, 876, 1075, 1130, 1247, 1343, 1450, 1537, 1833, 1909, 2248, 2362, 2515, 2668, 2850, 2940, 3400, 3587, 3789, 3919, 4477, 4624, 5242, 5440, 5654, 5916

Offset: 1

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)

Cf. A027424, A027426, A027430.

import Data.List (nub)
a027425 n = length $ nub [i*j*k | i <- [1..n], j <- [1..n], k <- [1..n]]
-- Reinhard Zumkeller, Jan 01 2012

f:=proc(n) local i,j,k,t1,t2; t1:={}; for i from 1 to n do for j from i to n do for k from j to n do t1:={op(t1),i*j*k}; od: od: od: t1:=convert(t1,list); nops(t1); end;

a[n_] := Reap[Do[Sow[i*j*k], {i, 1, n}, {j, i, n}, {k, j, n}]][[2, 1]] // Union // Length; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 30 2018 *)

pr(n)=my(v=List());for(i=1,n, for(j=i,n, listput(v, i*j))); Set(v)
a(n)=my(v=pr(n),u=v); for(i=2,n,u=Set(concat(u,v*i))); #u \\ Charles R Greathouse IV, Mar 04 2014

def A027425(n): return len({i*j*k for i in range(1,n+1) for j in range(1,i+1) for k in range(1,j+1)}) # Chai Wah Wu, Oct 16 2023

a(n) = A027426(n)-1. - T. D. Noe, Jan 16 2007

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}]

A100440 Number of distinct values of ij + jk + k*i with 1 <= i <= j <= k <= n.

Original entry on oeis.org

1, 4, 10, 20, 33, 50, 68, 93, 123, 154, 193, 233, 276, 325, 377, 434, 500, 568, 643, 720, 804, 885, 979, 1068, 1168, 1274, 1381, 1495, 1615, 1746, 1876, 2005, 2148, 2285, 2437, 2596, 2748, 2908, 3077, 3241, 3425, 3608, 3796, 3979, 4181, 4388, 4585, 4804, 5015, 5237

Offset: 1

N. J. A. Sloane, Nov 21 2004

nonn

a(n) <= A000292(n); a(n) = number of terms in n-th row of the triangle in A200741. - Reinhard Zumkeller, Nov 21 2011

David A. Corneth, Table of n, a(n) for n = 1..3000

Cf. A027430, A100439, A102533, A102534, A200737.

a100440 = length . a200741_row  -- Reinhard Zumkeller, Nov 21 2011

f:=proc(n) local i,j,k,t1; t1:={}; for i from 1 to n do for j from i to n do for k from j to n do t1:={op(t1),i*j+j*k+k*i}; od: od: od: t1:=convert(t1,list); nops(t1); end;

f[n_] := Length[ Union[ Flatten[ Table[i*j + j*k + k*i, {i, n}, {j, i, n}, {k, j, n}] ]]]; Table[ f[n], {n, 48}] (* Robert G. Wilson v, Dec 14 2004 *)

first(n) = {my(v = vector(3*n^2, i, oo), res = vector(n)); forvec(x = vector(3, i, [1,n]), c = x[1]*x[2] + x[1]*x[3] + x[2]*x[3]; v[c] = min(x[3],v[c]); , 1); for(i = 1, #v, if(v[i] < oo, res[v[i]]++)); for(i = 2, #res, res[i] += res[i-1]); res } \\ David A. Corneth, Mar 23 2021

from numba import njit
@njit()
def aupton(terms):
  aset, alst = set(), []
  for n in range(1, terms+1):
    for i in range(1, n+1):
      for j in range(i, n+1):
        aset.add(i*j + j*n + n*i)
    alst.append(len(aset))
  return alst
print(aupton(50)) # Michael S. Branicky, Mar 23 2021

More terms from Robert G. Wilson v, Dec 14 2004

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.

A027428 Number of distinct products ij with 1 <= i < j <= n. (Number of terms appearing more than once in a 1-to-n multiplication table.)

Original entry on oeis.org

0, 1, 3, 6, 10, 13, 19, 24, 31, 36, 46, 51, 63, 70, 78, 87, 103, 111, 129, 139, 150, 161, 183, 192, 210, 223, 239, 252, 280, 291, 321, 337, 354, 371, 390, 403, 439, 458, 478, 493, 533, 549, 591, 611, 631, 654, 700, 717, 752, 774, 800, 823, 875

Offset: 1

N. J. A. Sloane

Branden Aldridge, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe).

Cf. A027424, A027427, A027430.

import Data.List (nub)
a027428 n = length $ nub [i*j | j <- [2..n], i <- [1..j-1]]
-- Reinhard Zumkeller, Jan 01 2012

f:=proc(n) local i,j,t1,t2; t1:={}; for i from 1 to n-1 do for j from i+1 to n do t1:={op(t1),i*j}; od: od: t1:=convert(t1,list); nops(t1); end;

a[n_] := Table[i*j, {i, 1, n-1}, {j, i+1, n}] // Flatten // Union // Length; Table[ a[n] , {n, 1, 53}] (* Jean-François Alcover, Jan 31 2013 *)

def A027428(n): return len({i*j for i in range(1,n+1) for j in range(1,i)}) # Chai Wah Wu, Oct 13 2023

a(n) = A027427(n) - 1. - T. D. Noe, Jan 16 2007

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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A100448 Number of triples (i,j,k) with 1 <= i <= j < k <= n and gcd{i,j,k} = 1.

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A027425 Number of distinct products ijk with 1 <= i,j,k <= n.

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

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

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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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A100440 Number of distinct values of i*j + j*k + k*i with 1 <= i <= j <= k <= n.

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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.

A100440 Number of distinct values of ij + jk + k*i with 1 <= i <= j <= k <= n.

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.