A053042 -id:A053042 - cp's OEIS frontend

A055775 a(n) = floor(n^n / n!).

1, 1, 2, 4, 10, 26, 64, 163, 416, 1067, 2755, 7147, 18613, 48638, 127463, 334864, 881657, 2325750, 6145596, 16263866, 43099804, 114356611, 303761260, 807692034, 2149632061, 5726042115, 15264691107, 40722913454, 108713644516

Offset: 0

Henry Bottomley, Jul 12 2000

Stirling's approximation for n! suggests that this should be about e^n/sqrt(pi*2n). Bill Gosper has noted that e^n/sqrt(pi*(2n+1/3)) is significantly better.
n^n/n! = A001142(n)/A001142(n-1), where A001142(n) is product{k=0 to n} C(n,k) (where C() is a binomial coefficient). - Leroy Quet, May 01 2004
There are n^n distinct functions from [n] to [n] or sequences on n symbols of length n, the number of those sequences having n distinct symbols is n!. So the probability P(n) of bijection is n!/n^n. The expected value of the number of functions that we pick until we found a bijection is the reciprocal of P(n), or n^n/n!. - Washington Bomfim, Mar 05 2012

			a(5)=26 since 5^5=3125, 5!=120, 3125/120=26.0416666...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Washington Bomfim, A method to find bijections from a set of n integers to {0,1, ... ,n-1}
Eric Weisstein's World of Mathematics, Stirling's Approximation for n!

Cf. A073225, A094082, A053042, A036679, A061711, A152170, A209081, A208846, A208847.

[Floor((n^n)/Factorial(n)): n in [0..30]]; // Vincenzo Librandi, Aug 22 2011

Join[{1}, Table[Floor[n^n/n!], {n, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)

a(n)=n^n\n! \\ Charles R Greathouse IV, Apr 17 2012

a(n) = floor(A000312(n)/A000142(n)).

More terms from James Sellers, Jul 13 2000

A061711 a(n) = n^n * n!.

Original entry on oeis.org

1, 1, 8, 162, 6144, 375000, 33592320, 4150656720, 676457349120, 140587147048320, 36288000000000000, 11388728893445164800, 4270826380475341209600, 1886009588552176549862400, 968725766854884321342259200, 572622616354851562500000000000

Offset: 0

Lorenzo Fortunato (fortunat(AT)pd.infn.it), Jun 19 2001

a(n) is the product of first n terms of an arithmetic progression with first term n and common difference n. E.g. a(3) = 3*6*9 = 162. - Amarnath Murthy, Sep 20 2003

Product of the entries in the last column of an n X n square array whose elements are the numbers 1..n^2 listed in increasing order by rows. - Wesley Ivan Hurt, Mar 31 2025

			a(1) = 1^1 * 1! = 1;
a(2) = 2^2 * 2! = 8;
a(3) = 3^3 * 3! = 162.

Harry J. Smith, Table of n, a(n) for n = 0..100 (corrected by Michel Marcus, Jan 19 2019)

Cf. A036679, A053042, A055775. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009
Main diagonal of A131182.
Cf. A336765.

[Factorial(n)*n^n: n in [0..30]]; // G. C. Greubel, Nov 29 2022

Table[If[n == 0, 1, n^n] * n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 08 2018 *)

a(n) = n!*n^n; \\ Harry J. Smith, Jul 26 2009

from math import factorial
def A061711(n): return factorial(n)*n**n # Chai Wah Wu, Sep 03 2022

E.g.f.: sinh(n*x)^n. - Vaclav Kotesovec, Nov 05 2014
a(n) = [x^n] 1/(1 - n*x/(1 - n*x/(1 - 2*n*x/(1 - 2*n*x/(1 - 3*n*x/(1 - 3*n*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Sep 20 2017
Sum_{n>=1} 1/a(n) = A336765. - Amiram Eldar, Nov 20 2020
a(n) ~ exp(-n)*n^(2*n)*sqrt(2*n*Pi). - Peter Luschny, Jan 10 2022

A354888 a(n) = n! * Sum_{d|n} d^d / d!.

Original entry on oeis.org

1, 6, 33, 328, 3245, 52056, 828583, 17328256, 389416329, 10105386400, 285351587411, 8955841614336, 302881333613053, 11126513414294656, 437935136609883375, 18455736024587862016, 827240617573764860177, 39353706314004951028224

Offset: 1

Seiichi Manyama, Jun 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A327579, A354843, A354889.

a[n_] := n! * DivisorSum[n, #^#/#! &]; Array[a, 18] (* Amiram Eldar, Jun 10 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, d^d/d!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (k*x)^k/(k!*(1-x^k)))))

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

If p is prime, a(p) = p^p + p! = A053042(p).

A354890 a(n) = n! * Sum_{d|n} d^n / d!.

Original entry on oeis.org

1, 6, 33, 472, 3245, 157896, 828583, 132078976, 1578211209, 307174074400, 285351587411, 1835340563252736, 302881333613053, 11743240652094910336, 336123967242674523375, 149825956013958069846016, 827240617573764860177, 3551697093896307129060647424

Offset: 1

Seiichi Manyama, Jun 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..284

Cf. A053042, A354888, A354891, A354892.

a[n_] := n! * DivisorSum[n, #^n/#! &]; Array[a, 18] (* Amiram Eldar, Jun 10 2022 *)

PARI
```
a(n) = n!*sumdiv(n, d, d^n/d!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (k*x)^k/(k!*(1-(k*x)^k)))))

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

If p is prime, a(p) = p^p + p! = A053042(p).

A109663 Numbers k such that the sum of the digits of (k^k + k!) is divisible by k.

Original entry on oeis.org

1, 2, 3, 9, 15, 18, 27, 36, 51, 81, 93, 169, 181, 348, 444, 504, 528, 1881, 2031, 9843, 16479, 16685, 45435, 129056, 138510, 214008, 358326

Offset: 1

Ryan Propper, Aug 06 2005

The quotients are 2, 3, 2, 6, 6, 5, 8, 7, 6, 9, 9, 10, 10, 12, 12, 12, 12, 15, 15, 18, 19, 19, 21, 23, 22, 24, 25.

No more terms < 500000. - Lars Blomberg, Jul 05 2011

			The digits of 1881^1881 + 1881! sum to 28215 and 28215 is divisible by 1881, so 1881 is in the sequence.

Cf. A053042, A109662.

Do[s = n^n + n!; k = Plus @@ IntegerDigits[s]; If[Mod[k, n] == 0, Print[n]], {n, 1, 10000}]
Select[Range[360000],Divisible[Total[IntegerDigits[#^#+#!]],#]&] (* Harvey P. Dale, Dec 27 2018 *)

a(21)-a(27) from Lars Blomberg, Jul 05 2011

A301347 a(n) = n^(n-1) + (n-1)!.

Original entry on oeis.org

2, 3, 11, 70, 649, 7896, 118369, 2102192, 43087041, 1000362880, 25941053401, 743048287488, 23298564124081, 793721000274944, 29193013203681825, 1152922812281214976, 48661212798456756481, 2185911915426124627968, 104127356700284947260841, 5242880121645100408832000

Offset: 1

Seiichi Manyama, Mar 19 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A000142, A000169, A053042, A126130.

Array[#^(# - 1) + (# - 1)! &, 20] (* Michael De Vlieger, Mar 19 2018 *)

a(n) = n^(n-1) + (n-1)! \\ Altug Alkan, Mar 19 2018

a(n) = A053042(n)/n.

a(n) = A000169(n) + A000142(n-1).

cp's OEIS Frontend

A055775 a(n) = floor(n^n / n!).

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A061711 a(n) = n^n * n!.

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A354888 a(n) = n! * Sum_{d|n} d^d / d!.

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A354890 a(n) = n! * Sum_{d|n} d^n / d!.

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A109663 Numbers k such that the sum of the digits of (k^k + k!) is divisible by k.

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Mathematica

Extensions

A301347 a(n) = n^(n-1) + (n-1)!.

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Mathematica

PARI

Formula