A038507 -id:A038507 - cp's OEIS frontend

A277081 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged under the operation of replacing a permutation with its inverse.

1, 1, 1, 1, 1, 2, 1, 1, 4, 7, 8, 7, 4, 1, 1, 10, 52, 190, 546, 1302, 2660, 4754, 7535, 10692, 13672, 15820, 16604, 15820, 13672, 10692, 7535, 4754, 2660, 1302, 546, 190, 52, 10, 1, 1, 26, 372, 3822, 31306, 216086, 1300420, 6981650, 33992275, 151945820

Offset: 0

Christian Bean, Sep 28 2016

T(n,k) is the number of size k subsets of S_n that remain unchanged under the operation of replacing a permutation with its inverse.

			For n = 3 and k = 3 the subsets unchanged by inverse are {213,132,123}, {321,132,123}, {321,213,123}, {231,312,123}, {321,132,213}, {132,312,231},{213,312,231}, {321,231,312} hence T(3,3) = 8. (Here we are using the one-line notation for permutations, not the product of cycles form.)
Triangle starts:
1, 1;
1, 1;
1, 2, 1;
1, 4, 7, 8, 7, 4, 1;

Andrew Howroyd, Table of n, a(n) for n = 0..880 (rows 0..6)

Row lengths give A038507.

Cf. A000085.

\\ here b(n) is A000085(n).
b(n)={sum(k=0, n\2, n!/((n-2*k)!*2^k*k!))}
Row(n)={my(t=b(n)); vector(n!+1, k, k--; sum(i=0, k\2, binomial((n!-t)/2, i)*binomial(t, k-2*i)))}
{ for(n=0, 4, print(Row(n))) } \\ Andrew Howroyd, Feb 03 2021

T(n,k) = Sum( C((n! - I(n))/2, i)*C(I(n), k - 2*i) for i in [0..floor(k/2)]) where I(n) = A000085(n).

A301316 a(n) = ((n-1)! + 1) mod n^2.

Original entry on oeis.org

0, 2, 3, 7, 0, 13, 35, 49, 64, 81, 11, 1, 0, 57, 1, 1, 85, 1, 38, 1, 1, 133, 184, 1, 1, 521, 1, 1, 522, 1, 589, 1, 1, 885, 1, 1, 259, 381, 1, 1, 656, 1, 559, 1, 1, 553, 282, 1, 1, 1, 1, 1, 1802, 1, 1, 1, 1, 2553, 1593, 1, 3416, 993, 1, 1, 1, 1, 804

Offset: 1

Stanislav Sykora, Mar 18 2018

nonn

By definition, when n > 1, a(n) = 0 then n is a Wilson prime (A007540).
For a(n) to equal 1, (n-1)! must be divisible by n^2 which is the prevailing case for large n. For example, all n which are a product of more than two distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, and 3^2. Obviously, when a(n) = 1, then also A055976(n) = 1.
The cases of a(n) > 1 include, for example, all primes other than Wilson's and all numbers of the form n=2*p, where p is a prime.

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^2 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^2 = (1! +1) mod 4 = 2 mod 4 = 2.
((3-1)! + 1) mod 3^2 = (2! +1) mod 9 = 3 mod 9 = 3.
((4-1)! + 1) mod 4^2 = (3! +1) mod 16 = 7 mod 16 = 7.
((5-1)! + 1) mod 5^2 = (4! +1) mod 25 = 25 mod 25 = 0.
... (End)

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson prime

Cf. A000290, A038507, A007540, A055976, A301317.

List([1..60],n->(Factorial(n-1)+1) mod n^2); # Muniru A Asiru, Mar 20 2018

seq((factorial(n-1)+1) mod n^2,n=1..60); # Muniru A Asiru, Mar 20 2018

Array[Mod[(# - 1)! + 1, #^2] &, 67] (* Michael De Vlieger, Apr 21 2018 *)

a(n) = ((n-1)! + 1) % n^2; \\ Michel Marcus, Mar 18 2018

a(n) = ((n-1)! + 1) mod n^2. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000290(n). - Michel Marcus, Mar 20 2018

A301317 a(n) = (n-1)! + 1 mod n^3.

Original entry on oeis.org

0, 2, 3, 7, 25, 121, 35, 433, 226, 881, 495, 1, 676, 1233, 2701, 2049, 4420, 1, 4009, 1, 2647, 6425, 4945, 1, 626, 15393, 1, 1, 13137, 1, 21731, 1, 13069, 2041, 1, 1, 23532, 19153, 50194, 1, 14104, 1, 41237, 1, 1, 76729, 86433, 1, 1, 1, 78031, 1, 77645

Offset: 1

Stanislav Sykora, Mar 18 2018

nonn

There is no known number n > 1 for which a(n)=0.
For a(n) to equal 1, (n-1)! must be divisible by n^3 which tends to be the most frequent case for large n. For example, all n which are a product of three or more distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, 2^4, 3^2, and 5^2.
Obviously, when a(n) = 1, then also A055976(n) = 1 and A301316(n) = 1.
If n is prime, a(n) is divisible by n. - Robert Israel, Mar 20 2018

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^3 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^3 = (1! +1) mod 8 = 2 mod 8 = 2.
((3-1)! + 1) mod 3^3 = (2! +1) mod 27 = 3 mod 27 = 3.
((4-1)! + 1) mod 4^3 = (3! +1) mod 64 = 7 mod 64 = 7.
((5-1)! + 1) mod 5^3 = (4! +1) mod 125 = 25 mod 125 = 25.
... (End)

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson prime

Cf. A000578, A038507, A055976, A301316.

List([1..60],n->(Factorial(n-1)+1) mod n^3); # Muniru A Asiru, Mar 20 2018

seq((factorial(n-1)+1) mod n^3,n=1..60); # Muniru A Asiru, Mar 20 2018

Array[Mod[(# - 1)! + 1, #^3] &, 53] (* Michael De Vlieger, Mar 19 2018 *)

a(n) = ((n-1)! + 1) % n^3; \\ Michel Marcus, Mar 18 2018

a(n) = ((n-1)! + 1) mod n^3. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000578(n). - Michel Marcus, Mar 20 2018

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Original entry on oeis.org

3, 7, 8, 25, 26, 30, 121, 122, 126, 144, 721, 722, 726, 744, 840, 5041, 5042, 5046, 5064, 5160, 5760, 40321, 40322, 40326, 40344, 40440, 41040, 45360, 362881, 362882, 362886, 362904, 363000, 363600, 367920, 403200, 3628801, 3628802, 3628806, 3628824, 3628920, 3629520, 3633840, 3669120, 3991680

Offset: 1

Seiichi Manyama, Mar 23 2018

nonn

Numbers of the form i! + j! where i > j > 0. - Altug Alkan, Mar 23 2018

Primes in this sequence are A088332(n) for n > 1.

			    + |   1    2    6   24
  ----+--------------------
    1 |
    2 |   3;
    6 |   7,   8;
   24 |  25,  26,  30;
  120 | 121, 122, 126, 144;

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000142, A001048, A038507, A059590, A088332, A301593.

Union[Total/@Subsets[Range[10]!,{2}]] (* Harvey P. Dale, Aug 25 2020 *)

A358280 a(n) = Sum_{d|n} (d-1)!.

Original entry on oeis.org

1, 2, 3, 8, 25, 124, 721, 5048, 40323, 362906, 3628801, 39916930, 479001601, 6227021522, 87178291227, 1307674373048, 20922789888001, 355687428136444, 6402373705728001, 121645100409194912, 2432902008176640723, 51090942171713068802, 1124000727777607680001

Offset: 1

Seiichi Manyama, Nov 08 2022

Cf. A062363, A321875.

Cf. A038507, A358279.

a[n_] := DivisorSum[n, (# - 1)! &]; Array[a, 23] (* Amiram Eldar, Aug 30 2023 *)

PARI
```
a(n) = sumdiv(n, d, (d-1)!);
```

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, (k-1)!*x^k/(1-x^k)))

G.f.: Sum_{k>0} (k-1)! * x^k/(1 - x^k).

If p is prime, a(p) = 1 + (p-1)!.

A362777 Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 25, 49, 73, 97, 121, 241, 361, 481, 601, 721, 1441, 2161, 2881, 3601, 4321, 5041, 10081, 15121, 20161, 25201, 30241, 35281, 40321, 80641, 120961, 161281, 201601, 241921, 282241, 322561, 362881, 725761, 1088641, 1451521, 1814401, 2177281, 2540161, 2903041, 3265921

Offset: 1

Joe B. Stephen, May 03 2023

These numbers are used in a simple proof of the infinitude of the primes: n!*i + 1 and n!*j + 1 are coprime for 1 <= i < j <= n, so for any n we get n coprime integers (greater than 1) and hence we get at least n distinct primes.

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6 ...
  1    2
  2    3    5
  3    7   13   19
  4   25   49   73   97
  5  121  241  361  481  601
  6  721 1441 2161 2881 3601 4321
  ...

Cf. A362778, A362779.

Cf. A038507 (1st column), A188914 (right diagonal).

A366758 a(n) is the sum of the divisors of n!+1.

Original entry on oeis.org

3, 3, 4, 8, 31, 133, 832, 5113, 41044, 388800, 3958704, 39916802, 518682390, 6302045232, 90968651712, 1332614649600, 22844265373440, 356226551466344, 7504470340300800, 123358411682195904, 2432902126073962432, 52279222588118377280, 1175121515279802150144

Offset: 0

Sean A. Irvine, Oct 20 2023

nonn

			a(5) = 133 because the divisors of 5!+1 are {1, 11, 121}.

Cf. A038507, A000203, A064144, A062569, A366757.

a:=n->numtheory[sigma](n!+1):
seq(a(n), n=0..30);

DivisorSigma[1,Range[0,25]!+1] (* Paolo Xausa, Oct 21 2023 *)

from math import factorial
from sympy import divisor_sigma
def A366758(n): return divisor_sigma(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = sigma(n!+1) = A000203(A038507(n)).

A366760 a(n) = phi(n!+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 2, 6, 20, 110, 612, 4970, 39600, 337680, 3298900, 39916800, 442155168, 6151996372, 83387930692, 1282826630160, 19089488332800, 355148307803520, 5427568925856000, 119931789135468100, 2432901890279317572, 49902667163053013232, 1073067539495604750240

Offset: 0

Sean A. Irvine, Oct 20 2023

nonn

Cf. A038507, A000010, A048855, A366759.

EulerPhi[Range[0,25]!+1] (* Paolo Xausa, Oct 21 2023 *)

PARI
```
{a(n) = eulerphi(n!+1)}
```

from math import factorial
from sympy import totient
def A366760(n): return totient(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = A000010(A038507(n)).

A064011 Sum of distinct primes dividing n! + 1.

Original entry on oeis.org

2, 3, 7, 5, 11, 110, 71, 722, 359, 329902, 39916801, 2834342, 75024430, 3790360510, 46271879, 1059865, 1538931, 123611150, 1713311273363902, 117897322430, 2703876255255, 93799610095770191, 148139754736864717

Offset: 1

Jason Earls, Sep 07 2001

nonn

			a(6) = 110 since 6! + 1 = 721 = 7 * 103 and 7 + 103 = 110.

Amiram Eldar, Table of n, a(n) for n = 1..139 (terms 1..60 from Harry J. Smith)

Cf. A008472, A038507.

sopf[n_] := Plus @@ First @ Transpose @ FactorInteger[n]; sopf /@ Table[n! + 1 ,{n, 1, 23}] (* Amiram Eldar, Feb 03 2020 *)

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1], s=s+fac[i,1]); return(s);
for(n=1,23,print(sopf(n!+1)))

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ for (n=1, 60, write("b064011.txt", n, " ", sopf(n! + 1)) ) } \\ Harry J. Smith, Sep 06 2009

a(n) = A008472(A038507(n)). - Amiram Eldar, Feb 03 2020

A067241 Numbers k such that gcd((2*k)!+1, k!+1) > 1.

Original entry on oeis.org

3, 5, 8, 9, 21, 23, 33, 39, 51, 63, 65, 81, 89, 95, 99, 113, 131, 173, 183, 191, 209, 215, 221, 239, 245, 251, 261, 281, 285, 299, 303, 309, 315, 341, 345, 363, 369, 371, 393, 411, 419, 431, 443, 473, 495, 509, 525, 543, 545, 561, 575, 593, 645, 659, 683, 711

Offset: 1

Benoit Cloitre, Feb 20 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A038507, A127231.

q[k_] := !CoprimeQ[(2*k)!+1, k!+1]; Select[Range[720], q] (* Amiram Eldar, May 01 2025 *)

isok(k) = gcd((2*k)!+1, k!+1) > 1; \\ Amiram Eldar, May 01 2025

cp's OEIS Frontend

A277081 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged under the operation of replacing a permutation with its inverse.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A301316 a(n) = ((n-1)! + 1) mod n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A301317 a(n) = (n-1)! + 1 mod n^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A358280 a(n) = Sum_{d|n} (d-1)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A362777 Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A366758 a(n) is the sum of the divisors of n!+1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A366760 a(n) = phi(n!+1), where phi is Euler's totient function (A000010).

Views