A070863 -id:A070863 - cp's OEIS frontend

A060957 Number of different products (including the empty product) of any subset of {1, 2, 3, ..., n}.

1, 1, 2, 4, 8, 16, 26, 52, 88, 152, 238, 476, 648, 1296, 2016, 2984, 4232, 8464, 11360, 22720, 30544, 43744, 67072, 134144, 166336, 242752, 370992, 498144, 656832, 1313664, 1581312, 3162624, 3960384, 5517248, 8386080, 11111232, 13065792, 26131584, 39690432

Offset: 0

Jonas Wallgren, May 10 2001

a(n) <= 2*a(n-1), with equality iff n is prime or n = 4. - Martin Fuller, Jun 03 2006
a(n) = 2^k * b(n) where k is the number of primes p such that n/2 < p <= n, and b(n) is the number of different products of subsets of {1, 2, ..., n} that exclude these primes. - David Radcliffe, Feb 11 2019
Conjecture: Let p <= n be prime. If m and p^a*m are two such products, then so is p^k*m for all 0 < k < a. - Yan Sheng Ang, Feb 13 2020
a(n) is even for n > 1. Since k is a product implies that n!/k is a product, a(n) is odd implies that n! is a square, which is impossible for n > 1 because of the Bertrand's postulate: for n > 1, there is a prime p in the range (n/2, n], so p divides n! while p^2 does not. - Jianing Song, Sep 26 2022

			a(4) = 8: the subsets of {1, 2, 3, 4} are {}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}. The 16 numbers as the product are 1, 1, 2, 3, 4, 2, 3, 4, 6, 8, 12, 6, 8, 12, 24. There are only 8 distinct numbers: 1, 2, 3, 4, 6, 8, 12, 24.
a(6) = 26: the set {1, 2, 3, 4, 5, 6, 2*3, 2*4, 2*5, ..., 5*6, 2*3*4, 2*3*5, ..., 4*5*6, ..., ...2*3*4*5*6} contains 26 different values: {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, 36, 40, 48, 60, 72, 90, 120, 144, 180, 240, 360, 720}

Yan Sheng Ang, Table of n, a(n) for n = 0..68 (first 50 terms from David Radcliffe)

Cf. A070861, A070863, A255937, A307105.

s:= proc(n) option remember; `if`(n=0, {1},
      map(x-> [x, x*n][], s(n-1)))
    end:
a:= n-> nops(s(n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 25 2016

(* Script not convenient for n > 24 *) a[n_] := Times @@@ Subsets[Range[n]] // Union // Length; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 24}] (* Jean-François Alcover, Feb 02 2015 *)
s[n_] := s[n] = If[n == 0, {1}, Map[Function[x, {x, x*n}], s[n-1]]  // Flatten // Union]; a[n_] := Length[s[n]]; Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 30}] (* Jean-François Alcover, Nov 01 2016, after Alois P. Heinz *)

from functools import cache
@cache
def s(n): return {1} if n == 0 else s(n-1) | set(x*n for x in s(n-1))
def a(n): return len(s(n))
print([a(n) for n in range(30)]) # Michael S. Branicky, Jul 31 2022 after Alois P. Heinz

More terms from Lior Manor, May 26 2002
a(26)-a(32) from Giovanni Resta, Feb 14 2006
More terms from Martin Fuller, Jun 03 2006
a(0)=1 and a(37)-a(38) from Alois P. Heinz, Aug 25 2016

A070861 Triangle of all possible distinct numbers obtained as a product of distinct numbers from 1..n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, 36, 40, 48, 60, 72, 90, 120, 144, 180, 240, 360, 720, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42

Offset: 0

Amarnath Murthy, May 16 2002

Factorials are a subsequence (A000142). - Reinhard Zumkeller, Jul 02 2011

More generally, all sequences of positive integers are subsequences. - Charles R Greathouse IV, Mar 06 2017

			Triangle begins:
  1;
  1;
  1, 2;
  1, 2, 3, 6;
  1, 2, 3, 4, 6, 8, 12, 24;
  ...

Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Cf. A000142, A060957.
Row sums give A070863.
Row products give A283261.

a070861 n = a070861_list !! (n-1)
a070861_list = concat a070861_tabf
a070861_tabf = [1] : f 2 [1] where
   f n ps = ps' : f (n+1) ps' where ps' = m ps $ map (n*) ps
   m []         ys = ys
   m xs'@(x:xs) ys'@(y:ys)
       | x < y     = x : m xs ys'
       | x == y    = x : m xs ys
       | otherwise = y : m xs' ys
b070861 = bFile' "A070861" (concat $ take 20 a070861_tabf) 1
-- Reinhard Zumkeller, Jul 02 2011

T:= proc(n) option remember; `if`(n=0, 1,
      sort([map(x-> [x, x*n][], {T(n-1)})[]])[])
    end:
seq(T(n), n=0..7);  # Alois P. Heinz, Aug 01 2022

row[n_] := Times @@@ Subsets[Range[n]] // Flatten // Union; Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 02 2015 *)

row(n)=my(v=[2..n]); Set(vector(2^(n-1),i, factorback(vecextract(v,i-1)))) \\ Charles R Greathouse IV, Mar 06 2017

T(n,A060957(n)) = A000142(n) = n!. - Alois P. Heinz, Aug 01 2022

Corrected and extended by Lior Manor, May 23 2002

Row n=0 prepended by Alois P. Heinz, Aug 01 2022

A283261 Product of the different products of subsets of the set of numbers from 1 to n.

Original entry on oeis.org

1, 1, 2, 36, 331776, 42998169600000000, 13974055172471046820331520000000000000, 1833132881579690383668380351534446872452674453158326975200092938148249600000000000000000000000000

Offset: 0

Jaroslav Krizek, Mar 04 2017

nonn

Product of numbers in n-th row of A070861.

			Rows with subsets of the sets of numbers from 1 to n:
  {},
  {}, {1};
  {}, {1}, {2}, {1, 2};
  {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3};
  ...
Rows with the products of elements of these subsets:
  1;
  1, 1;
  1, 1, 2, 2;
  1, 1, 2, 3, 2, 3, 6, 6;
  ...
Rows with the different products of elements of these subsets:
  1;
  1;
  1, 2;
  1, 2, 3, 6;
  ...
a(0) = 1, a(1) = (1), a(2) = (1*2) = 2, a(3) = (1*2*3*6) = 36, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..10

Cf. A000005, A000142, A027423, A060957, A070863, A052295, A263292.

b:= proc(n) option remember; `if`(n=0, {1},
      map(x-> [x, x*n][], b(n-1)))
    end:
a:= n-> mul(i, i=b(n)):
seq(a(n), n=0..7);  # Alois P. Heinz, Aug 01 2022

Table[Times @@ Union@ Map[Times @@ # &, Subsets@ Range@ n], {n, 7}] (* Michael De Vlieger, Mar 05 2017 *)

a(n)=my(v=[2..n]); factorback(Set(vector(2^(n-1),i, factorback(vecextract(v,i-1))))) \\ Charles R Greathouse IV, Mar 06 2017

a(n) <= n!^((A000005(n!))/2) = n!^(A027423(n)/2). - David A. Corneth, Mar 05 2017

a(n) = n!^(A263292(n)). - David A. Corneth, Mar 06 2017

a(0)=1 prepended by Alois P. Heinz, Aug 01 2022

cp's OEIS Frontend

A060957 Number of different products (including the empty product) of any subset of {1, 2, 3, ..., n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A070861 Triangle of all possible distinct numbers obtained as a product of distinct numbers from 1..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A283261 Product of the different products of subsets of the set of numbers from 1 to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions