A070861 -id:A070861 - cp's OEIS frontend

A070863 Sum of numbers in n-th row of A070861.

1, 1, 3, 12, 60, 360, 2268, 18144, 152544, 1471008, 14963328, 179559936, 2156963328, 30197486592, 426680825088, 6448500066288, 103658110188528, 1865845983393504, 33623263082197152, 672465261643943040, 13457623759369050240, 283699943666342340480, 6265115909183775026880

Offset: 0

Amarnath Murthy, May 16 2002

nonn

Cf. A070861, A060957.

b:= proc(n) option remember; `if`(n=0, {1},
      map(x-> [x, x*n][], b(n-1)))
    end:
a:= n-> add(i, i=b(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 01 2022

row[n_] := Times @@@ Subsets[Range[n]] // Flatten // Union; a[n_] := Total[row[n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 19}] (* Jean-François Alcover, Feb 02 2015 *)

Corrected and extended by Lior Manor, May 26 2002

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

A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each 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, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

This sequence is the same as A070861 for the first 38 terms, but differs thereafter.

			First few rows are:
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;
...

Andrew Howroyd, Table of n, a(n) for n = 0..1979 (rows 0..12)

Cf. A027423 (row lengths), A062569 (row sums), A070861.

[Divisors(Factorial(n)): n in [0..10]]; // Vincenzo Librandi, Jun 19 2015

Flatten[Table[Divisors[n!],{n,6}]]  (* Harvey P. Dale, Mar 13 2011 *)

tabf(nn) = for (n=0, nn, print(divisors(n!))); \\ Michel Marcus, Jun 19 2015

a(0)=1 prepended by Andrew Howroyd, Jan 26 2022

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

Original entry on oeis.org

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

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

A070863 Sum of numbers in n-th row of A070861.

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A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each n.

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

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

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Maple

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