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 1-4 of 4 results.

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

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

A284988 Number of distinct products i*j*k*l*m for 1 <= i <= j <= k <= l <= m <= n.

Original entry on oeis.org

1, 6, 21, 36, 91, 126, 266, 336, 441, 546, 994, 1120, 1890, 2184, 2562, 2856, 4482, 4932, 7392, 8052, 9042, 10032, 14377, 15092, 17237, 18887, 20812, 22297, 30635, 31856, 42783, 45240, 49023, 52806, 57707, 59436, 77623, 83083, 89180, 92365, 118188, 122248, 154188
Offset: 1

Views

Author

Seiichi Manyama, Apr 07 2017

Keywords

Links

Crossrefs

Cf. A027424, A027425, A100437.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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