A036679 -id:A036679 - cp's OEIS frontend

A162604 Primes of the form k^k-k!-9.

223, 16736887, 104857597567097991823359991, 11756638905368320778615010897169507936008298389569527

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

nonn

The values of A036679(k)-9 at indices k=4, 8, 20, 34, ... - R. J. Mathar, Jul 16 2009

[a: n in [3..100] | IsPrime(a) where a is n^n-Factorial(n)-9]; // Vincenzo Librandi, Aug 02 2017

Select[Table[n^n-n!-9,{n,3,50}],PrimeQ] (* Harvey P. Dale, Jul 30 2017 *)

A344116 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.

Original entry on oeis.org

1, 3, 14, 7, 58, 506, 15, 242, 4060, 65512, 31, 994, 32618, 1048336, 33554312, 63, 4034, 261604, 16775656, 1073740024, 68719476016, 127, 16258, 2095346, 268427056, 34359721568, 4398046495984, 562949953416272, 255, 65282, 16771420, 4294926472, 1099511501776, 281474976519136, 72057594037786816, 18446744073709511296

Offset: 1

Mohammad K. Azarian, Jun 07 2021

			For T(2,2), the number of relations is 2^4 and the number of onto functions is 2, so 2^4 - 2 = 14.
Triangle T(n,k) begins:
   1
   3     14
   7     58      506
  15    242     4060      65512
  31    994    32618    1048336    33554312

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A101030, A131689, A199656, A344110, A344112, A344113, A344115.

TableForm[Table[2^(n*k) - Sum[Binomial[k, k - i] (k - i)^n*(-1)^i, {i, 0, k}], {n, 5}, {k, n}]]

T(n,k) = 2^(n*k) - k!*stirling(n, k, 2); \\ Michel Marcus, Jun 26 2021

T(n,k) = 2^(n*k) - k!*Stirling2(n,k).

T(n,k) = A344110(n,k) - A131689(n,k).

A109662 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, 14, 15, 29, 33, 45, 81, 102, 105, 126, 142, 157, 288, 414, 1184, 2133, 10449, 16369, 17221, 46524, 214179, 216741

Offset: 1

Ryan Propper, Aug 06 2005

The quotients are 0, 1, 1, 5, 5, 6, 7, 6, 8, 9, 9, 9, 9, 10, 10, 11, 12, 14, 15, 18, 19, 19, 21, 24, 24.
No more terms < 500000. - Lars Blomberg, Jul 05 2011
a(26) > 595261. - J.W.L. (Jan) Eerland, Nov 05 2024

			The digits of 414^414 - 414! sum to 4968 and 4968 is divisible by 414, so 414 is in the sequence.

Cf. A036679, A109663.

Do[s = n^n - n!; k = Plus @@ IntegerDigits[s]; If[Mod[k, n] == 0, Print[n]], {n, 1, 10000}]

Terms a(20)-a(25) from Lars Blomberg, Jul 05 2011

A114846 Numbers of the form p^p - p!, where p is a prime.

Original entry on oeis.org

2, 21, 3005, 818503, 285271753811, 302868879571453, 827239906198908668177, 1978419534015213180291979, 20880467973995895295470056270567, 2567686153152369372568088512776472510483469, 17069174130715013119956465106241589193164734431

Offset: 1

Giovanni Teofilatto, Feb 20 2006

Except for the term 2, p^p - p! is equal to the sum of p distinct primes.

Cf. A036679.

Table[Prime[n]^Prime[n] - Prime[n]!, {n, 1, 11}] (* Stefan Steinerberger, Feb 21 2006 *)
Table[p^p-p!,{p,Prime[Range[15]]}] (* Harvey P. Dale, Feb 03 2015 *)

More terms from Stefan Steinerberger, Feb 21 2006

A162605 Primes of the form k^k-k!+2.

Original entry on oeis.org

2, 23, 387057611, 7596040312163297274222442578036920783684137730466972969164241364503367485389656454146145546668114950475963285686810192201741431

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

nonn

The values of A036679(k)+2 at indices k=1, 3, 9, 69, ... - R. J. Mathar, Jul 16 2009

No further terms up to k=1000. - Harvey P. Dale, Apr 26 2022

Cf. A162603, A162604

f[n_]:=n^n-n!+2; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,4*5!}]; lst
Select[Table[k^k-k!+2,{k,100}],PrimeQ] (* Harvey P. Dale, Apr 26 2022 *)

A344919 a(n) = n^n - n*(n + 1) / 2.

Original entry on oeis.org

1, 0, 1, 21, 246, 3110, 46635, 823515, 16777180, 387420444, 9999999945, 285311670545, 8916100448178, 302875106592162, 11112006825557911, 437893890380859255, 18446744073709551480, 827240261886336764024, 39346408075296537575253, 1978419655660313589123789, 104857599999999999999999790

Offset: 0

Peter Luschny, Jun 19 2021

Cf. A000217, A000312, A036679, A058126.

Join[{1},Table[n^n-(n(n+1))/2,{n,20}]] (* Harvey P. Dale, Sep 07 2024 *)

def A344919(n):
    return (n**n)-(n*(n+1)//2) # Karl-Heinz Hofmann, Jun 19 2021

a(n) = Product_{j=1..n} n - Sum_{j=1..n} j.

a(n) = A000312(n) - A000217(n).

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

Original entry on oeis.org

0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800

Offset: 1

Mohammad K. Azarian, Aug 28 2021

The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).

			For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
       k=1  k=2   k=3   k=4    k=5     k=6
  n=1:  0    0    0     0      0       0
  n=2:       2    3     4      5       6
  n=3:            21    40     65      96
  n=4:                  232    505     936
  n=5:                         3005    7056
  n=6:                                 45936

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.

A347034 := proc(n,k)
    k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023

Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten

T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021

T(n,k) = k^n - k!/(k - n)!, k>=n.

T(n,n) = A036679(n).

A359950 a(n) is the greatest prime factor of n^n - n!.

Original entry on oeis.org

2, 7, 29, 601, 29, 116929, 11887, 4778489, 82207, 296987, 2767, 464089, 36922117, 71722471217, 10219277051, 9406703479, 2040247819, 122450719, 1265072927, 18353142818474353, 21514105057, 46999724987, 29693667067, 5684341885088084044195811037649, 692132186353, 12114317049616531

Offset: 2

Sebastian F. Orellana, Jan 19 2023

nonn

			a(5) = greatest prime factor of 5^5 - 5! = greatest prime factor of 3125 - 120 = greatest prime factor of 3005 = 3005/5 = 601.

Amiram Eldar, Table of n, a(n) for n = 2..106

Cf. A006530, A020639, A036679, A126130.

Table[Max[First/@FactorInteger[n^n-n!]],{n,2,27}] (* Stefano Spezia, Jan 22 2023 *)

a(n) = vecmax(factor(n^n - n!)[,1]); \\ Michel Marcus, Jan 22 2023

a(n) = A006530(A036679(n)) = A006530(n*A126130(n-1)).

More terms from Michel Marcus, Jan 22 2023

A068983 a(n) = Sum_{k=0..n} (k^k-k!).

Original entry on oeis.org

0, 2, 23, 255, 3260, 49196, 867699, 17604595, 404662204, 10401033404, 295672787215, 9211294233871, 312080173805324, 11423999821072140, 449316582527563515, 18896039733447227131, 846135945932355895308, 40192537618855187742732, 2018612071634068368034711

Offset: 1

Darrell Minor, Apr 02 2002

a(n) = number of non-injective functions [k]->[k] for 1<=k<=n.

			a(4) = 255 because (1^1-1!)+(2^2-2!)+(3^3-3!)+(4^4-4!) = 255.

Cf. A003422, A062970, A036679.

Accumulate[Table[n^n-n!,{n,20}]] (* Harvey P. Dale, Aug 21 2011 *)

a(n) = Sum_{k=0..n} (k^k-k!).

a(n) = A062970(n) - A003422(n+1). - Alois P. Heinz, Aug 10 2021

A162606 Primes of the form k^k-k!+6.

Original entry on oeis.org

3011, 818509, 827239906198908668183, 3877924263464448622408024944643162574255380546190593937673127995497376385062869

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

nonn

The values of A036679(k)+6 at indices k=5, 7, 17, 47, 97, 1073, ... - R. J. Mathar, Jul 16 2009

Cf. A162603, A162604, A162605

f[n_]:=n^n-n!+6; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,p]],{n,3*5!}]; lst
Select[Table[n^n-n!+6,{n,50}],PrimeQ] (* Harvey P. Dale, Sep 22 2019 *)

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A162604 Primes of the form k^k-k!-9.

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A344116 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.

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A109662 Numbers k such that the sum of the digits of (k^k - k!) is divisible by k.

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A114846 Numbers of the form p^p - p!, where p is a prime.

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A162605 Primes of the form k^k-k!+2.

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A344919 a(n) = n^n - n*(n + 1) / 2.

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A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

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A359950 a(n) is the greatest prime factor of n^n - n!.

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