A126130 -id:A126130 - cp's OEIS frontend

A036679 a(n) = n^n - n!.

0, 0, 2, 21, 232, 3005, 45936, 818503, 16736896, 387057609, 9996371200, 285271753811, 8915621446656, 302868879571453, 11111919647266816, 437892582706491375, 18446723150919663616, 827239906198908668177, 39346401672922831847424, 1978419534015213180291979

Offset: 0

N. J. A. Sloane, G. L. Honaker, Jr.

a(n) = |non-injective functions [n]->[n]| = |non-surjective functions [n]->[n]|.
Fit a polynomial f of degree n-1 to the first n n-th powers of nonnegative integers. Then a(n) = f(n). - Franklin T. Adams-Watters, Dec 28 2006
n^n > n! for n >= 3. [Mitrinovic]

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 193, 3.1.22.

T. D. Noe and Vincenzo Librandi, Table of n, a(n) for n = 0..300 [T. D. Noe computed terms 0-50, May 11 2007; Vincenzo Librandi computed the first 300 terms, Aug 22 2011]
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141. See Corollary 2.3(iii).

Cf. A126130, diagonal of A101030.

[(n^n-Factorial(n)): n in [0..20] ]; // Vincenzo Librandi, Aug 22 2011

Join[{0}, Table[n^n - n!, {n, 20}]] (* Harvey P. Dale, Oct 11 2011 *)

a(n)=n^n-n! \\ Charles R Greathouse IV, Aug 22 2011

from math import factorial
def a(n): return n**n - factorial(n)
print([a(n) for n in range(20)]) # Michael S. Branicky, Aug 10 2021

E.g.f.: 1/(1-T(x))-1/(1-x) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Dec 10 2012

A226805 P_n(n+1) where P_n(x) is the polynomial of degree n-1 which satisfies P_n(i) = i^i for i = 1,...,n.

Original entry on oeis.org

1, 7, 70, 877, 13316, 237799, 4885980, 113566121, 2946476764, 84417530491, 2647176188372, 90183424037293, 3316840864313484, 130985236211745959, 5528094465439087876, 248308899812296990033, 11827417687501017074876, 595470029978391175571923

Offset: 1

José María Grau Ribas, Jun 18 2013

nonn

			P_3(x) = 18 - 27*x + 10*x^2; a(3) = P_3(3+1) = 70.

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A140119, A140118, A126130, A000312.

P[n_][x_] = Sum[a[i]*x^i, {i, 0, n - 1}];ecu[n_] := Table[P[n][i] == i^i, {i, 1, n}];PP[n_][x_] := P[n][x] /. Solve[ecu[n]][[1]];Table[PP[i][i + 1], {i, 1, 22}]
a[n_] := InterpolatingPolynomial[Table[{i, i^i}, {i, n}], n+1]; Array[a, 20] (* Giovanni Resta, Jun 18 2013 *)

a(n)=subst(polinterpolate(vector(n,i,i^i)),'x,n+1) \\ Charles R Greathouse IV, Nov 19 2013

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).

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

cp's OEIS Frontend

A036679 a(n) = n^n - n!.

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A226805 P_n(n+1) where P_n(x) is the polynomial of degree n-1 which satisfies P_n(i) = i^i for i = 1,...,n.

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

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Author

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Programs

Mathematica

PARI

Formula

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

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Mathematica

PARI

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