A060462 -id:A060462 - cp's OEIS frontend

A326180 Number of maximal subsets of {1..n} containing n whose product is divisible by their sum.

0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 1, 16, 1, 1, 1, 27, 1

Offset: 0

Gus Wiseman, Jun 13 2019

			The a(6) = 3, a(10) = 11, and a(12) = 16 subsets:
  {1,3,5,6}    {1,2,4,5,6,7,10}      {1,2,3,4,5,6,7,8,12}
  {1,2,3,4,6}  {1,2,3,4,5,7,8,10}    {1,3,4,5,6,7,8,10,12}
  {2,3,4,5,6}  {1,2,3,4,6,7,9,10}    {1,3,4,6,7,8,9,10,12}
               {1,2,3,5,6,7,8,10}    {1,3,4,5,6,8,10,11,12}
               {1,2,3,5,7,8,9,10}    {1,2,3,4,5,6,8,9,10,12}
               {1,2,5,6,7,8,9,10}    {1,2,3,4,6,7,8,9,11,12}
               {1,3,4,5,6,7,9,10}    {1,2,3,5,6,7,8,9,10,12}
               {1,3,4,6,7,8,9,10}    {1,2,3,5,6,7,8,9,11,12}
               {1,4,5,6,7,8,9,10}    {1,3,4,5,6,7,8,9,11,12}
               {1,2,3,4,5,6,8,9,10}  {1,2,3,4,6,7,8,10,11,12}
               {2,3,4,5,6,7,8,9,10}  {1,2,3,4,6,8,9,10,11,12}
                                     {1,3,5,6,7,8,9,10,11,12}
                                     {1,2,3,4,5,6,7,9,10,11,12}
                                     {1,2,3,4,5,7,8,9,10,11,12}
                                     {1,2,4,5,6,7,8,9,10,11,12}
                                     {2,3,4,5,6,7,8,9,10,11,12}

Cf. A053632, A057567, A057568, A059529, A060462, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326179.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n],{1,n}],MemberQ[#,n]&&Divisible[Times@@#,Plus@@#]&]]],{n,0,10}]

a(A060462(n)) = 1.

A118742 Numbers n for which the expression n!/(n+1) is an integer.

Original entry on oeis.org

0, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 22 2006

nonn

Also set of all n>=0, excluding 3, for which n+1 is composite. [Proof: (i) If n+1 is prime, there cannot be any factor in n! to cancel the n+1 in the denominator of the expression. (ii) If n+1=composite=a*b, a2, (n+1)!/(n+1)^2 = 1*2*..*a*...*(2a)*..*a^2/a^4 in which factors also cancel.] - R. J. Mathar, Nov 22 2006

			n=5 5!/(5+1)= 5*4*3*2*1/6 = 20.

Cf. A060462, A120416, A270441.

Essentially the same as A072668.

P:=proc(n) local i,j; for i from 0 by 1 to n do j:=i!/(i+1); if trunc(j)=j then print(i); fi; od; end: P(200);

Select[Range[0,100],IntegerQ[#!/(#+1)]&] (* Harvey P. Dale, Aug 24 2014 *)

a(n) = A002808(n+1)-1 for n>=1. - R. J. Mathar, Nov 22 2006

Corrected (39 inserted) by Harvey P. Dale, Aug 24 2014

A166460 Numbers k such that k + (-1)^k is not prime.

Original entry on oeis.org

0, 1, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94

Offset: 1

Juri-Stepan Gerasimov, Oct 14 2009

This is the complement of A068499 (except that both include 1 as a term).
From Don Reble, Aug 31 2021: (Start)
Proof for all k except 0, 1, 3 with cases
(i)   If k is odd and >=5, then k+1 = 2*x, 2 < x < k, k! = k*...*x*...*2*1
        A068499: k+1 divides k!            : absent
        A166460: k-1 is even and composite : present
(ii)  If k is even and k+1 is prime,
        A068499: k+1 does not divide k!    : present
        A166460: k+1 is prime              : absent
(iii) If k is even and k+1 = p^2 is the square of a (odd) prime, then k+1 >= 3p, k > 2p.
        A068499: k! = k*...*2p*...*p*...*1;
                 k+1 divides k!            : absent
        A166460: k+1 is composite          : present
(iv)  If k is even and k+1 is composite but not the square of a prime, then there are two distinct factors x*y = k+1:
                 3 <= x < y = (k+1)/x < k.
        A068499: k! = k*...*y*...*x*...*1:
                 k+1 divides k!            : absent
        A166460: k+1 is composite          : present
(End)

			0 + (-1)^0 = 1 is not prime, which adds 0 to the sequence.
5 + (-1)^5 = 4 is not prime, which adds 5 to the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002808, A072668, A141468, A060462, A118742, A068499.

Select[Range[0, 94], ! PrimeQ[# + (-1)^#] &] (* Michael De Vlieger, Sep 08 2021 *)

from sympy import composite
def A166460(n): return composite(n-1)-1 if n>2 else n-1 # Chai Wah Wu, Aug 27 2024

a(n) = A002808(n-1)-1 for n>2. - Chai Wah Wu, Aug 27 2024

0 added by R. J. Mathar, Oct 21 2009

A126328 Rounded value of n!/(n(n+1)/2); A000142(n)/A000217(n).

Original entry on oeis.org

1, 1, 1, 2, 8, 34, 180, 1120, 8064, 65978, 604800, 6141046, 68428800, 830269440, 10897286400, 153844043294, 2324754432000, 37440781904842, 640237370572800, 11585247657984000, 221172909834240000, 4442690623626907826

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 09 2007

nonn

This is the value that comes up when (all positive integers n and below multiplied together) is divided by (all positive integers n and below added together). Some terms need to be rounded, but for some terms, n! is divisible by n(n+1)/2. See A060462 for these numbers.

			a(6) = 34 because 6! = 720 and 6(6+1)/2 = 21. 720/21 = 34.2857... rounded to 34.

Cf. A000142, A000217, A060462.

Cf. A061370. [From R. J. Mathar, Dec 13 2008]

Table[Round[n!/(n(n+1)/2)],{n,22}] (* James C. McMahon, Dec 25 2024 *)

A291551 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Product_{i=1..j} s_i.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 0, 262, 0, 10226, 43964, 139484, 0, 13936472, 59652396, 301235944, 1915640632, 7969506364, 0

Offset: 0

Seiichi Manyama, Aug 26 2017

a(n) = 0 if n+1 does not divide 2*(n-1)!. This implies that a(p-1) = 0 for p > 2 prime. - Chai Wah Wu, Aug 26 2017

			a(15) = 26: [[10, 15, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 1, 9, 12], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 7, 1, 9, 4, 8], [10, 15, 5, 6, 12, 2, 14, 11, 13, 3, 9, 4, 1, 7, 8], [10, 15, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 3, 12], [10, 15, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 12, 3], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 7, 1, 9, 4, 8], [10, 15, 5, 6, 14, 13, 12, 3, 2, 11, 9, 4, 1, 7, 8], [15, 10, 5, 6, 4, 8, 2, 14, 11, 13, 3, 7, 1, 9, 12], [15, 10, 5, 6, 12, 2, 14, 11, 13, 3, 7, 1, 9, 4, 8], [15, 10, 5, 6, 12, 2, 14, 11, 13, 3, 9, 4, 1, 7, 8], [15, 10, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 2, 7, 8, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 7, 2, 8, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 7, 8, 2, 11, 9, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 3, 12], [15, 10, 5, 6, 14, 13, 9, 8, 11, 7, 2, 4, 1, 12, 3], [15, 10, 5, 6, 14, 13, 12, 3, 2, 11, 7, 1, 9, 4, 8], [15, 10, 5, 6, 14, 13, 12, 3, 2, 11, 9, 4, 1, 7, 8]].

Cf. A060462, A126328.

def search(a, prod, sum, size, num)
  if num == size + 1
    @cnt += 1
  else
    (1..size).each{|i|
      p, s = prod * i, sum + i
      if a[i - 1] == 0 && p % s == 0
        a[i - 1] = 1
        search(a, p, s, size, num + 1)
        a[i - 1] = 0
      end
    }
  end
end
def A(n)
  a = [0] * n
  @cnt = 0
  search(a, 1, 0, n, 1)
  @cnt
end
def A291551(n)
  (0..n).map{|i| A(i)}
end
p A291551(20)

a(26)-a(28) from Alois P. Heinz, Aug 26 2017

A309355 Even numbers k such that k! is divisible by k*(k+1)/2.

Original entry on oeis.org

8, 14, 20, 24, 26, 32, 34, 38, 44, 48, 50, 54, 56, 62, 64, 68, 74, 76, 80, 84, 86, 90, 92, 94, 98, 104, 110, 114, 116, 118, 120, 122, 124, 128, 132, 134, 140, 142, 144, 146, 152, 154, 158, 160, 164, 168, 170, 174, 176, 182, 184, 186, 188, 194, 200, 202, 204, 206

Offset: 1

Gerhard Palme, Jul 25 2019

nonn

Even terms in A060462.
And A071904 are the successors of a(n).
Even numbers that are not a prime - 1. That is, even numbers not in A006093. - Terry D. Grant, Oct 31 2020

			8! = 40320 is divisible by 8*9/2 = 36.
14! is divisible by 14*15/2.

J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.
Die WURZEL - Zeitschrift für Mathematik, 53. Jahrgang, Juli 2019, S. 171, WURZEL-Aufgabe 2019-36 von Gerhard Dietel, Regensburg.

Cf. A060462, A071904.
Essentially the same as A186193.
Cf. A006093.

[k: k in [2..250]|IsEven(k) and Factorial(k) mod Binomial(k+1,2) eq 0]; // Marius A. Burtea, Jul 28 2019

Complement[Table[2 n, {n, 1, 103}], Table[EulerPhi[Prime[n]], {n, 1, 103}]] (* Terry D. Grant, Oct 31 2020 *)

forcomposite(c=4,10^3,if(c%2==1,print1(c-1,", "))); \\ Joerg Arndt, Jul 25 2019

from sympy import primepi
def A309355(n):
    if n == 1: return 8
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m-1 # Chai Wah Wu, Aug 02 2024

a(n) = A071904(n) - 1.

A343506 Numbers k such that the largest digit in the factorial base expansion of 1/k is 1.

Original entry on oeis.org

1, 2, 6, 20, 24, 120, 630, 720, 4480, 5040, 36288, 40320, 362880, 3326400, 3628800, 39916800

Offset: 1

Rémy Sigrist, Apr 17 2021

Equivalently these are the numbers k such that A299020(k) = 1 or A343505(k) = 1.
This sequence is infinite as it contains:
- the factorial numbers (A000142),
- 1/(1/A060462(k)! + 1/(A060462(k)-1)!) for k > 2,
- 1/(1/A120416(k)! + 1/(A120416(k)-1)! + 1/(A120416(k)-2)!) for k > 0.

			The first terms, alongside the factorial base expansion of their inverse, are:
  n   a(n)     1/a(n) in factorial base
  --  -------  ------------------------
   1        1  1
   2        2  0.1
   3        6  0.0 1
   4       20  0.0 0 1 1
   5       24  0.0 0 1
   6      120  0.0 0 0 1
   7      630  0.0 0 0 0 1 1
   8      720  0.0 0 0 0 1
   9     4480  0.0 0 0 0 0 1 1
  10     5040  0.0 0 0 0 0 1
  11    36288  0.0 0 0 0 0 0 1 1
  12    40320  0.0 0 0 0 0 0 1
  13   362880  0.0 0 0 0 0 0 0 1
  14  3326400  0.0 0 0 0 0 0 0 0 1 1
  15  3628800  0.0 0 0 0 0 0 0 0 1

Cf. A000142, A060462, A120416, A294168, A299020, A343505.

Cf. A333402 (decimal).

is(n) = my (f=1/n); for (r=2, oo, if (f==0, return (1), floor(f)>1, return (0), f=frac(f)*r))

cp's OEIS Frontend

A326180 Number of maximal subsets of {1..n} containing n whose product is divisible by their sum.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A118742 Numbers n for which the expression n!/(n+1) is an integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A166460 Numbers k such that k + (-1)^k is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A126328 Rounded value of n!/(n(n+1)/2); A000142(n)/A000217(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A291551 Number of permutations s_1,s_2,...,s_n of 1,2,...,n such that for all j=1,2,...,n, Sum_{i=1..j} s_i divides Product_{i=1..j} s_i.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Ruby

Extensions

A309355 Even numbers k such that k! is divisible by k*(k+1)/2.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A343506 Numbers k such that the largest digit in the factorial base expansion of 1/k is 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI