A110713 -id:A110713 - cp's OEIS frontend

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.

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

Seiichi Manyama, Dec 31 2018

			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, ...

Seiichi Manyama, Antidiagonals n = 1..25, flattened

Columns 1-5 give A001477, A027424, A027425, A100437, A284988

Main diagonal gives A110713.

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

A345882 Number of numbers expressible as b(1)b(2)...*b(n) with 1 <= b(i) <= i for each i.

Original entry on oeis.org

1, 2, 5, 11, 30, 64, 178, 382, 758, 1367, 3620, 7193, 19707, 40867, 75706, 130017, 339506, 667390, 1824656, 3724917, 6785689, 11545898, 30099090, 58833294, 105348580, 176098677, 282847446, 438090287, 1095200628, 2057512312, 5494259815, 10925293558, 19311381148

Offset: 1

David Galvin, Sep 16 2021

On replacing * with +, one gets A000124.

In other words, take n! = 1*2*3*...*n and replace any factor by any smaller number. a(n) is the number of different numbers that can be obtained. If b(i) is required to be a divisor of i, we get A027423. - N. J. A. Sloane, Sep 18 2021

			For n=3, b(1) must equal 1, b(2) can be 1 or 2, and b(3) can be 1, 2 or 3. This gives 3!=6 possible products: 1*1*1=1, 1*2*1=2, 1*1*2=2, 1*1*3=3, 1*2*2=4 and 1*2*3=6. Since 1*2*1=1*1*2, this process yields 5 distinct numbers, so a(3)=5.

Cf. A000124, A027423, A110713, A347685 (first differences), A347686.

list[1] := {1};
list[n_] := list[n] = DeleteDuplicates[Flatten[Table[i*list[n - 1], {i, 1, n}]]];
a[n_] := a[n] = Length[list[n]]; Table[a[n], {n, 1, 10}]

a(n) = my(l = List()); forvec(x = vector(n, i, [1, i]), listput(l, prod(i = 1, n, x[i])), 1); listsort(l, 1); #l \\ David A. Corneth, Sep 18 2021

def A345882set(n):
    if n == 1:
        return {1}
    else:
        s = A345882set(n-1)
        c = set(s)
        for x in s:
            for i in range(2,n+1):
                c.add(i*x)
        return c
def A345882(n): return len(A345882set(n)) # Chai Wah Wu, Sep 19 2021

a(26)-a(28) from Chai Wah Wu, Sep 19 2021

a(29)-a(33) from Martin Ehrenstein, Sep 22 2021

A160375 Given n, let S denote the set of numbers c_1c_2...*c_n where 1<=c_1<=c_2<=...<=c_n<=n; a(n) = number of members of S that have a unique representation of this form.

Original entry on oeis.org

1, 3, 10, 16, 61, 81, 337, 477, 601, 901, 4291, 5798, 27314, 33671, 45732, 59397, 299745, 421363, 2090647, 2739022, 4597263, 5401826, 27510715, 23666955

Offset: 1

Mats Granvik, May 11 2009

Number of combinations as in A001700.
From David A. Corneth, Sep 26 2016: (Start)
a(n + 1) / a(n) is fairly large if n + 1 is prime; for the given data, it's at least three. In the other cases it's less than 2.
Let p be a distinct product as described in the name. We look at the factors rather than the result. For n = 4, we see the product p = 1*2*3*3.
Let F(p) be a vector of size n which counts the frequency F_e of each e where 1 <= e <= n. For n = 4 and the product we find (1,1,2,0).
For n = 6, we can put the following restrictions on a vector F(p) = (f_1, f_2, f_3, f_4, f_5, f_6): Trivially, f_e >= 0, f_1+f_2+...+f_6 = 6.
Furthermore,
f_2 * f_3 = 0, as 2*3 = 1*6 and 1<=n=6 and 6<=n=6, so if f_2, f_3 > 0, the value of the product isn't unique, contradiction;
f_2 < 2, 2*2 = 1*4;
f_3 * f_4 = 0 as 3*4 = 2*6. (End)

			a(3) = 10 because there are 10 numbers that can be written as such a product in exactly one way:
1*1*1 = 1
1*1*2 = 2
1*1*3 = 3
1*2*2 = 4
1*2*3 = 6
2*2*2 = 8
1*3*3 = 9
2*2*3 = 12
2*3*3 = 18
3*3*3 = 27
There are 25 possible products of the numbers 1,2,3,4 (see A110713), but 9 of those products can be attained in multiple ways (e.g., 1*2*2*4 = 1*1*4*4), so a(4) = 25-9 = 16.

David A. Corneth, PARI program
Gerhard Kirchner, Theory and algorithm

Cf. A001700, A110713.

Table[Count[Split@ Sort@ Map[Times @@ # &, Union@ Map[Sort, Tuples[Range@ n, n]]], w_ /; Length@ w == 1], {n, 8}] (* Michael De Vlieger, Sep 26 2016 *)

a(7)-a(13) from Nathaniel Johnston, Nov 29 2010
a(14)-a(24) from Gerhard Kirchner, Aug 30 2016
Definition edited by N. J. A. Sloane, Sep 27 2016

A321164 Sum of distinct products b_1b_2...*b_n where 1<=b_i<=n.

Original entry on oeis.org

1, 7, 90, 1441, 38325, 916714, 37830100, 1194352181, 45241845825, 1951734678972, 147430334155104, 5664495439368403, 528359397843879784, 30798685780200874044, 1774211038440007650672, 99969666291681633988821, 13249072759407029981640765, 847264606120975715873578180

Offset: 1

Seiichi Manyama, Jan 10 2019

nonn

			a(2) = 1 + 2 + 4 = 7.
a(3) = 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 27 = 90.

Main diagonal of A321163.

Cf. A110713.

a:= proc(m) option remember; local b; b:=
      proc(n) option remember; `if`(n=0, {1},
        map(x-> seq(x*i, i=1..m), b(n-1)))
      end; forget(b); add(i, i=b(m))
    end:
seq(a(n), n=1..12);  # Alois P. Heinz, Jan 11 2019

a[m_] := a[m] = Module[{b}, b[n_] := b[n] = If[n==0, {1}, Map[Table[# i, {i, 1, m}]&, b[n-1]]] // Flatten // Union; b[m] // Total];
Array[a, 12] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

a(15)-a(18) from Alois P. Heinz, Jan 11 2019

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A345882 Number of numbers expressible as b(1)*b(2)*...*b(n) with 1 <= b(i) <= i for each i.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A160375 Given n, let S denote the set of numbers c_1*c_2*...*c_n where 1<=c_1<=c_2<=...<=c_n<=n; a(n) = number of members of S that have a unique representation of this form.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A321164 Sum of distinct products b_1*b_2*...*b_n where 1<=b_i<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions

A345882 Number of numbers expressible as b(1)b(2)...*b(n) with 1 <= b(i) <= i for each i.

A160375 Given n, let S denote the set of numbers c_1c_2...*c_n where 1<=c_1<=c_2<=...<=c_n<=n; a(n) = number of members of S that have a unique representation of this form.

A321164 Sum of distinct products b_1b_2...*b_n where 1<=b_i<=n.