cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A100435 Number of distinct products i*j*k for 1 <= i <= j < k <= n.

Original entry on oeis.org

0, 1, 4, 9, 18, 26, 44, 57, 76, 93, 135, 153, 212, 245, 282, 317, 414, 452, 575, 624, 690, 759, 935, 986, 1103, 1196, 1297, 1378, 1645, 1716, 2024, 2136, 2279, 2427, 2597, 2687, 3110, 3292, 3483, 3606, 4123, 4262, 4837, 5026, 5227, 5485, 6168, 6318, 6725
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Crossrefs

Cf. A027428, A027425, A027430, A100436.

Programs

  • Maple
    f:=proc(n) local i,j,k,t1; t1:={}; for i from 1 to n-1 do for j from i to n-1 do for k from j+1 to n do t1:={op(t1),i*j*k}; od: od: od: t1:=convert(t1,list); nops(t1); end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ i*j*k, {i, n}, {j, i, n}, {k, j + 1, n}] ]]]; Table[ f[n], {n, 49}] (* Robert G. Wilson v, Dec 14 2004 *)
  • Python
    def A100435(n): return len({i*j*k for i in range(1,n+1) for j in range(1,i) for k in range(1,j+1)}) # Chai Wah Wu, Oct 16 2023

Extensions

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

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

Views

Author

Olivier Gérard

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000292, A100448, A027430, A015631.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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))
  • Python
    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

Formula

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

A100436 Number of distinct products i*j*k for 1 <= i < j <= k <= n.

Original entry on oeis.org

0, 1, 4, 10, 20, 27, 46, 61, 84, 101, 147, 163, 226, 256, 292, 331, 434, 472, 601, 655, 719, 785, 968, 1016, 1143, 1233, 1346, 1433, 1713, 1778, 2099, 2219, 2363, 2509, 2677, 2763, 3202, 3381, 3573, 3690, 4223, 4360, 4951, 5149, 5347, 5598, 6298, 6449
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Crossrefs

Cf. A027425, A027430, A100435.

Programs

  • Maple
    f:=proc(n) local i,j,k,t1; t1:={}; for i from 1 to n-1 do for j from i+1 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;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ i*j*k, {i, n}, {j, i + 1, n}, {k, j, n}]]]]; Table[ f[n], {n, 48}] (* Robert G. Wilson v, Dec 14 2004 *)
  • Python
    def A100436(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)}) # Chai Wah Wu, Oct 16 2023

Extensions

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

A100437 Number of distinct products i*j*k*l for 1 <= i <= j <= k <= l <= n.

Original entry on oeis.org

1, 5, 15, 25, 55, 75, 140, 175, 225, 275, 448, 504, 770, 882, 1022, 1134, 1626, 1782, 2460, 2670, 2970, 3270, 4345, 4565, 5135, 5585, 6100, 6505, 8338, 8679, 10927, 11525, 12393, 13261, 14345, 14787, 18187, 19344, 20618, 21346, 25823, 26698
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Links

Crossrefs

Cf. A027425, A027430, A100435, A100436, A100438.

Programs

  • Maple
    f:=proc(n) local i,j,k,l,t1; t1:={}; for i from 1 to n do for j from i to n do for k from j to n do for l from k to n do t1:={op(t1),i*j*k*l}; od: od: od: od: t1:=convert(t1,list); nops(t1); end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ i*j*k*l, {i, n}, {j, i, n}, {k, j, n}, {l, k, n}] ]]]; Table[ f[n], {n, 45}] (* Robert G. Wilson v, Dec 14 2004 *)
  • PARI
    pr(n)=my(v=List());for(i=1,n, for(j=i,n, listput(v, i*j))); Set(v)
a(n)=my(u=List(),v=pr(n)); for(i=1,#v,for(j=i,#v,listput(u,v[i]*v[j]))); #Set(u) \\ Charles R Greathouse IV, Mar 04 2014

Extensions

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

A027427 Number of distinct products ij with 0 <= i < j <= n.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 14, 20, 25, 32, 37, 47, 52, 64, 71, 79, 88, 104, 112, 130, 140, 151, 162, 184, 193, 211, 224, 240, 253, 281, 292, 322, 338, 355, 372, 391, 404, 440, 459, 479, 494, 534, 550, 592, 612, 632, 655, 701, 718, 753, 775, 801, 824, 876
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A027430, etc.
Cf. A027384, A027428, A027429.

Programs

  • Haskell
    import Data.List (nub)
a027427 n = length $ nub [i*j | j <- [1..n], i <- [0..j-1]]
-- Reinhard Zumkeller, Jan 01 2012
  • Maple
    A027427 := proc(n)
    local L, i, j ;
    L := {};
    for i from 0 to n do
        for j from i+1 to n do
            L := L union {i*j};
        end do:
    end do:
    nops(L);
end proc:  # R. J. Mathar, Jun 09 2016
  • Mathematica
    a[n_] := Table[i*j, {i, 0, n}, {j, i+1, n}] // Flatten // Union // Length;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 03 2018 *)
  • Python
    def A027427(n): return 1+len({i*j for i in range(1,n+1) for j in range(1,i)}) if n else 0 # Chai Wah Wu, Oct 13 2023

Formula

a(n) = A027428(n)+1. - T. D. Noe, Jan 16 2007

A027429 Number of distinct products ijk with 0 <= i < j < k <= n.

Original entry on oeis.org

0, 0, 1, 2, 5, 11, 17, 30, 43, 61, 76, 112, 127, 178, 207, 239, 275, 362, 397, 508, 555, 614, 678, 839, 884, 1005, 1093, 1199, 1278, 1530, 1591, 1882, 1999, 2134, 2276, 2433, 2519, 2922, 3097, 3279, 3392, 3885, 4015, 4564, 4751, 4939, 5187, 5841, 5988, 6423
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			a(3) = 2 (0 and 6 being the only products) and a(4) = 5 (with products 0, 6, 8, 12 and 24).

Links

Crossrefs

Cf. A027425, A027426, A027427, A027430.

Programs

  • Haskell
    import Data.List (nub)
a027429 n = length $ nub [i*j*k | k<-[2..n], j<-[1..k-1], i<-[0..j-1]]
-- Reinhard Zumkeller, Jan 01 2012
  • Mathematica
    nn=50; prod=Table[0, {1+nn^3}]; t=Table[Do[prod[[1+i*j*k]]=1, {i,0,n}, {j,i+1,n}, {k,j+1,n}]; Count[Take[prod,1+n^3],1], {n,0,nn}] (* T. D. Noe, Jan 16 2007 *)
  • Python
    from itertools import combinations as C
def a(n): return len(set(i*j*k for i, j, k in C(range(n+1), 3)))
print([a(n) for n in range(50)]) # Michael S. Branicky, May 28 2021

Formula

a(n) = A027430(n) + 1. - T. D. Noe, Jan 16 2007

Extensions

Corrected by T. D. Noe, Jan 16 2007

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

Views

Author

Robert Price, May 09 2013

Keywords

Comments

This sequence is in reply to an extension request made in A100449.
Note that gcd(0,m) = m for any m.

Links

Crossrefs

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

Programs

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

A100439 Number of distinct values of i*j + j*k + k*i with 1 <= i

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 31, 48, 70, 96, 123, 161, 197, 240, 293, 340, 394, 460, 524, 596, 670, 744, 819, 918, 1016, 1112, 1217, 1322, 1429, 1561, 1679, 1798, 1943, 2072, 2218, 2379, 2518, 2669, 2838, 3009, 3178, 3361, 3536, 3713, 3924, 4120, 4304, 4522, 4727
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Crossrefs

Cf. A027430, A100440, A102533, A102534.

Programs

  • Maple
    f:=proc(n) local i,j,k,t1; t1:={}; for i from 1 to n-2 do for j from i+1 to n-1 do for k from j+1 to n do t1:={op(t1),i*j+j*k+k*i}; od: od: od: t1:=convert(t1,list); nops(t1); end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[i*j + j*k + k*i, {i, n}, {j, i + 1, n}, {k, j + 1, n}] ]]]; Table[ f[n], {n, 49}] (* Robert G. Wilson v, Dec 14 2004 *)

Extensions

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

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

Views

Author

Robert Price, May 09 2013

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

A100438 Number of distinct products i*j*k*l for 1 <= i < j < k < l <= n.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 29, 50, 79, 111, 186, 219, 345, 428, 513, 610, 884, 991, 1387, 1535, 1742, 1994, 2671, 2833, 3319, 3719, 4154, 4474, 5751, 5985, 7575, 8121, 8803, 9593, 10401, 10785, 13303, 14371, 15414, 15988, 19379, 20089, 24103, 25237, 26369
Offset: 1

Views

Author

N. J. A. Sloane, Nov 21 2004

Keywords

Crossrefs

Cf. A027425, A027430, A100435, A100436, A100437.

Programs

  • Maple
    f:=proc(n) local i,j,k,l,t1; t1:={}; for i from 1 to n-3 do for j from i+1 to n-2 do for k from j+1 to n-1 do for l from k+1 to n do t1:={op(t1),i*j*k*l}; od: od: od: od: t1:=convert(t1,list); nops(t1); end;
  • Mathematica
    f[n_] := Length[ Union[ Flatten[ Table[ i*j*k*l, {i, n}, {j, i + 1, n}, {k, j + 1, n}, {l, k + 1, n}]]]]; Table[ f[n], {n, 45}] (* Robert G. Wilson v, Dec 14 2004 *)
  • Python
    def A100438(n): return len({i*j*k*l for i in range(1,n+1) for j in range(1,i) for k in range(1,j) for l in range(1,k)}) # Chai Wah Wu, Oct 16 2023

Extensions

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