A060957 -id:A060957 - cp's OEIS frontend

A092965 Greatest prime arising as the product of numbers chosen from among the first n numbers + 1.

2, 3, 7, 13, 61, 241, 2521, 20161, 72577, 604801, 39916801, 59875201, 3113510401, 17435658241, 186810624001, 10461394944001, 118562476032001, 246245142528001, 24329020081766401, 304112751022080001

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

There are a maximum of 2^n numbers which arise as the products of the subsets of the first n natural numbers. The actual number is smaller because of repetitions. Then a(n) = the greatest prime obtained on adding 1 to each of these numbers.

Different from A089136 (see the comments there).

			a(5) = 61 = 3*4*5 + 1. 5! + 1, 4!+ 1, are composite and 2*4*5 + 1 = 41 <61, etc.

Martin Fuller, Table of n, a(n) for n = 1..200

Cf. A060957, A092967.

Do[l = Map[Times @@ #&, Subsets[Range[n]]]; Print[Max[Select[Map[ #+1&, l], PrimeQ]]], {n, 20}] (* Ryan Propper, Aug 13 2005 *)
f[n_] := Max@ Select[ Union[ Times @@@ Subsets@ Range@ n] + 1, PrimeQ]; Array[f, 20] (* Robert G. Wilson v, Nov 13 2014 *)

More terms from Ryan Propper, Aug 13 2005

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

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

Original entry on oeis.org

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

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

A092967 Largest prime of the form a squarefree number + 1 where the prime divisors of the squarefree number are < n.

Original entry on oeis.org

2, 3, 7, 7, 31, 31, 211, 211, 211, 211, 2311, 2311, 6007, 6007, 6007, 6007, 102103, 102103, 3233231, 3233231, 3233231, 3233231, 17160991

Offset: 1

Amarnath Murthy, Mar 26 2004

nonn

Conjecture: a(n)-1 has prime(n)-1 divisors. Subsidiary sequence: Number of primes of the form 2*p*q*r*...+1 where p, q, r, etc. are distinct odd primes < n.

			a(13) = 6007 = 2*3*7*11*13 + 1, as 2*5*7*11*13 + 1, etc. are composite.

Cf. A092965, A060957.

Mathematica
```
<Ryan Propper, Aug 13 2005 *)
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More terms from Ryan Propper, Aug 13 2005

A307105 Number of rational numbers which can be constructed from the set of integers between 1 and n, through a combination of multiplication and division.

Original entry on oeis.org

1, 1, 3, 9, 21, 63, 117, 351, 621, 1161, 2043, 6129, 8631, 25893, 45135, 71685, 102285, 306855, 420309, 1260927, 1755513, 2671299, 4571073, 13713219, 17156853, 25778169, 43930755, 59315085, 80765235, 242295705, 295267275, 885801825

Offset: 0

Brian Barsotti, Jul 07 2019

nonn

This sequence can contain only odd terms, because apart from 1, for every term x/y there is always the corresponding terms y/x. - Giovanni Resta, Jul 07 2019
a(n) <= 3*a(n-1), with equality iff n is prime. - Yan Sheng Ang, Feb 13 2020
Conjecture: Let p <= n be prime. If m and p^a*m are two such rationals, then so is p^k*m for all 0 < k < a. - Yan Sheng Ang, Feb 13 2020

			a(2) = 3 because {1,2} can create {1/2, 1, 2}.
a(3) = 9 because {1,2,3} can create {1/6, 1/3, 1/2, 2/3, 1, 3/2, 2, 3, 6}.
a(4) = 21 because {1,2,3,4} can create {1/24, 1/12, 1/8, 1/6, 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 2, 8/3, 3, 4, 6, 8, 12, 24}.

Yan Sheng Ang, Table of n, a(n) for n = 0..51

Cf. A018805, A060957.

s:= proc(n) option remember; `if`(n=0, {1},
      map(x-> [x, x*n, x/n][], s(n-1)))
    end:
a:= n-> nops(s(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 29 2019

L={}; s={1}; Do[s = Union[s, s/k, s*k]; AppendTo[L, Length@ s], {k, 13}]; L (* Giovanni Resta, Jul 07 2019 *)

a(p) = 3 * a(p-1), for p prime. - Giovanni Resta, Jul 07 2019

a(9)-a(31) from Giovanni Resta, Jul 07 2019

A255937 Number of distinct products of distinct factorials up to n!.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 28, 56, 108, 204, 332, 664, 1114, 2228, 4078, 7018, 11402, 22804, 40638, 81276, 140490, 230328, 391544, 783088, 1287034, 2273676, 3903626, 6837760, 10368184, 20736368, 34081198, 68162396

Offset: 0

Charles R Greathouse IV, Mar 11 2015

			a(3) = |{1!, 2!, 3!, 2!*3!}| = |{1, 2, 6, 12}| = 4.

Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.

Cf. A058295, A000142, A001013, A060957.

s:= proc(n) option remember; (f-> `if`(n=0, {f},
      map(x-> [x, x*f][], s(n-1))))(n!)
    end:
a:= n-> nops(s(n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 16 2015

a[n_] := a[n] = If[n == 0, 1, If[PrimeQ[n], 2 a[n-1], Times @@@ ((Subsets[Range[n]] // Rest) /. k_Integer -> k!) // Union // Length]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, May 01 2022 *)

a(n)=my(v=[1],N=n!); for(k=2,n-1, v=Set(concat(v,v*k!))); #v + sum(i=1,#v, !setsearch(v,N*v[i]))

Erdős and Graham prove that log a(n) ~ n log log n/log n.

a(p) = 2*a(p-1) for prime p. - Jon E. Schoenfield, Apr 01 2015

More terms from Alois P. Heinz, Mar 16 2015

a(31) (=2*a(30)) from Jon E. Schoenfield, Apr 01 2015

A255962 Number of repeating products of any subset of {1, 2, 3, ..., n}.

Original entry on oeis.org

0, 1, 3, 7, 15, 37, 75, 167, 359, 785, 1571, 3447, 6895, 14367, 29783, 61303, 122607, 250783, 501567, 1018031, 2053407, 4127231, 8254463, 16610879, 33311679, 66737871, 133719583, 267778623, 535557247, 1072160511, 2144321023, 4291006911, 8584417343, 17171483103, 34348627135, 68706410943

Offset: 1

Derek Orr, Mar 11 2015

nonn

			a(3) = (number of possible subsets of {1,2,3}) - |{1, 2, 3, 1*2, 1*3, 2*3, 1*2*3}| = 2^3-1 - |{1,2,3,6}| = 3. Equivalently, there are three repeating products (2, 3, and 6) so a(3) = 3.

Cf. A000225, A060957.

(* Script not convenient for n > 24 *) f[n_] := Block[{lst = Times @@@ Subsets[Range@ n, n]}, 2^n - 1 - Length@ Select[Tally@ lst, Last@ # > 1 &]]; Array[f, 16] (* Michael De Vlieger, Mar 13 2015 *)

a(n) = 2^n - 1 - A060957(n) = A000225(n) - A060957(n).

cp's OEIS Frontend

A092965 Greatest prime arising as the product of numbers chosen from among the first n numbers + 1.

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

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

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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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A092967 Largest prime of the form a squarefree number + 1 where the prime divisors of the squarefree number are < n.

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A307105 Number of rational numbers which can be constructed from the set of integers between 1 and n, through a combination of multiplication and division.

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Maple

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A255937 Number of distinct products of distinct factorials up to n!.

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