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.

Showing 1-10 of 12 results. Next

A027430 Number of distinct products i*j*k with 1 <= i < j < k <= n.

Original entry on oeis.org

0, 0, 1, 4, 10, 16, 29, 42, 60, 75, 111, 126, 177, 206, 238, 274, 361, 396, 507, 554, 613, 677, 838, 883, 1004, 1092, 1198, 1277, 1529, 1590, 1881, 1998, 2133, 2275, 2432, 2518, 2921, 3096, 3278, 3391, 3884, 4014, 4563, 4750, 4938, 5186, 5840, 5987, 6422, 6652
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Amarnath Murthy, Generalization of partition function introducing Smarandache Factor Partitions, Smarandache Notions Journal, 1-2-3, Vol. 11, 2000.

Links

Crossrefs

Cf. A000292, A027425, A088434, A100435, A100436.
Number of terms in row n of A083507.
Cf. A027429, A027428.

Programs

  • Haskell
    import Data.List (nub)
a027430 n = length $ nub [i*j*k | k<-[3..n], j<-[2..k-1], i<-[1..j-1]]
-- Reinhard Zumkeller, Jan 01 2012
  • Mathematica
    nn = 50;
prod = Table[0, {1 + nn^3}];
a[1] = 0;
a[n_] := (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] - 1);
Array[a, nn] (* Jean-François Alcover, Jul 31 2018, after T. D. Noe *)
  • PARI
    \\ See PARI link. David A. Corneth, Jul 31 2018
  • Python
    def A027430(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)}) # Chai Wah Wu, Oct 16 2023

Formula

a(n) = A027429(n)-1. - T. D. Noe, Jan 16 2007
a(n) <= A000292(n - 2). - David A. Corneth, Jul 31 2018

Extensions

Corrected by David Wasserman, Nov 18 2004

A027384 Number of distinct products i*j with 0 <= i, j <= n.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 19, 26, 31, 37, 43, 54, 60, 73, 81, 90, 98, 115, 124, 143, 153, 165, 177, 200, 210, 226, 240, 255, 268, 297, 309, 340, 355, 373, 391, 411, 424, 461, 481, 502, 518, 559, 576, 619, 639, 660, 684, 731, 748, 779, 801, 828, 851, 904, 926, 957, 979, 1009, 1039
Offset: 0

Views

Author

Fred Schwab (fschwab(AT)nrao.edu)

Keywords

Comments

a(n) = A027420(n,0) = A027420(n,n). - Reinhard Zumkeller, May 02 2014

Links

Crossrefs

Equals A027424 + 1, n>0.
Cf. A027417, A027427, A027425.

Programs

  • Haskell
    import Data.List (nub)
a027384 n = length $ nub [i*j | i <- [0..n], j <- [0..n]]
-- Reinhard Zumkeller, Jan 01 2012
  • Maple
    A027384 := proc(n)
    local L,i,j ;
    L := {};
    for i from 0 to n do
        for j from i to n do
            L := L union {i*j};
        end do:
    end do:
    nops(L);
end proc: # R. J. Mathar, May 06 2016
  • Mathematica
    u = {}; Table[u = Union[u, n*Range[0, n]]; Length[u], {n, 0, 100}] (* T. D. Noe, Jan 07 2012 *)
  • PARI
    a(n) = {my(s=Set()); for (i=0, n, s = setunion(s, Set(vector(n+1, k, i*(k-1))));); #s;} \\ Michel Marcus, Jan 01 2019
  • Python
    def A027384(n): return len({i*j for i in range(1,n+1) for j in range(1,i+1)})+1 # Chai Wah Wu, Oct 13 2023

Formula

For prime p, a(p) = a(p - 1) + p. - David A. Corneth, Jan 01 2019

A027426 Number of distinct products ijk with 0 <= i,j,k <= n.

Original entry on oeis.org

1, 2, 5, 11, 17, 31, 41, 66, 81, 101, 121, 174, 195, 267, 302, 344, 379, 493, 537, 679, 733, 805, 877, 1076, 1131, 1248, 1344, 1451, 1538, 1834, 1910, 2249, 2363, 2516, 2669, 2851, 2941, 3401, 3588, 3790, 3920, 4478, 4625, 5243, 5441, 5655, 5917, 6647, 6799, 7197
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A027384, A027429.

Programs

  • Haskell
    import Data.List (nub)
a027426 n = length $ nub [i*j*k | i <- [0..n], j <- [0..n], k <- [0..n]]
-- Reinhard Zumkeller, Jan 01 2012
  • Maple
    a:=proc(n): nops({seq(seq(seq(i*j*k,k=0..j),j=0..i),i=0..n)}) end: seq(a(n),n=0..50); # Emeric Deutsch, Jan 25 2007
  • Mathematica
    a[n_] := Table[i*j*k, {i, 0, n}, {j, i, n}, {k, j, n}] // Flatten // Union // Length; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 30 2018 *)
  • PARI
    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+1 \\ Charles R Greathouse IV, Mar 04 2014
  • Python
    from itertools import combinations_with_replacement as mc
def a(n): return len(set(i*j*k for i, j, k in mc(range(n+1), 3)))
print([a(n) for n in range(50)]) # Michael S. Branicky, May 28 2021

Formula

a(n) = A027425(n) + 1. - T. D. Noe, Jan 16 2007

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

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

A110713 a(n) is the number of distinct products b_1*b_2*...*b_n where 1 <= b_i <= n.

Original entry on oeis.org

1, 3, 10, 25, 91, 196, 750, 1485, 3025, 5566, 23387, 38402, 163268, 284376, 500004, 795549, 3575781, 5657839, 25413850, 40027130, 66010230, 105164280, 490429875, 713491350, 1232253906
Offset: 1

Views

Author

Jonas Wallgren, Sep 15 2005

Keywords

Comments

If * is changed to + the result is A002061. - Michel Marcus and David Galvin, Sep 19 2021

Examples

			a(2) = A027424(2) = 3.
a(3) = A027425(3) = 10.
a(4) = A100437(4) = 25.

Links

Crossrefs

Main diagonal of A322967.
Cf. A110713, A027424, A027425, A100437, A345882.

Programs

  • PARI
    a(n) = my(l = List()); forvec(x = vector(n, i, [1,n]), listput(l, prod(i = 1, n, x[i])), 1); listsort(l, 1); #l \\ David A. Corneth, Jan 02 2019
  • Python
    from math import prod
from itertools import combinations_with_replacement
def A110713(n): return len({prod(d) for d in combinations_with_replacement(list(range(1,n+1)),n)}) # Chai Wah Wu, Sep 19 2021

Extensions

a(10)-a(15) from Donovan Johnson, Dec 08 2009
a(16)-a(25) from Gerhard Kirchner, Dec 07 2015

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

A322967 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of distinct products Product_{j=1..k} b_j with 1 <= b_j<= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 9, 5, 1, 6, 15, 16, 14, 6, 1, 7, 21, 25, 30, 18, 7, 1, 8, 28, 36, 55, 40, 25, 8, 1, 9, 36, 49, 91, 75, 65, 30, 9, 1, 10, 45, 64, 140, 126, 140, 80, 36, 10, 1, 11, 55, 81, 204, 196, 266, 175, 100, 42, 11
Offset: 1

Views

Author

Seiichi Manyama, Dec 31 2018

Keywords

Examples

			In case of (n,k) = (3,2):
  | 1  2  3
--+--------
1 | 1, 2, 3
2 | 2, 4, 6
3 | 3, 6, 9
Distinct products are 1,2,3,4,6,9. So A(3,2) = 6.
Square array begins:
   1,  1,   1,   1,   1,   1,    1,    1,    1, ...
   2,  3,   4,   5,   6,   7,    8,    9,   10, ...
   3,  6,  10,  15,  21,  28,   36,   45,   55, ...
   4,  9,  16,  25,  36,  49,   64,   81,  100, ...
   5, 14,  30,  55,  91, 140,  204,  285,  385, ...
   6, 18,  40,  75, 126, 196,  288,  405,  550, ...
   7, 25,  65, 140, 266, 462,  750, 1155, 1705, ...
   8, 30,  80, 175, 336, 588,  960, 1485, 2200, ...
   9, 36, 100, 225, 441, 784, 1296, 2025, 3025, ...

Links

Crossrefs

Columns 1-5 give A001477, A027424, A027425, A100437, A284988
Main diagonal gives A110713.

Programs

  • Mathematica
    Table[Length@ Union@ Flatten[TensorProduct @@ ConstantArray[Range@ #, k]] &[n - k + 1], {n, 11}, {k, n, 1, -1}] // Flatten (* Michael De Vlieger, Jan 01 2019 *)

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
Showing 1-10 of 12 results. Next

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