A174379 -id:A174379 - cp's OEIS frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

-1, -1, -1, 17, 399, 7849, 162287, 3667649, 91171007, 2486784401, 74062575399, 2395420006033, 83695120256591, 3143661612445145, 126375169532421599, 5415486851106043649, 246486713303685957375, 11877172892329028459041, 604107995057426434824791

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

From Mathew Englander, Jul 07 2020: (Start)
All a(n) are odd and for n even, a(n) == 3 (mod 4); for n odd and n != 1, a(n) == 1 (mod 4).
The correspondence between n and a(n) when considered mod 6 is as follows: for n == 0, 1, 2, or 3, a(n) == 5; for n == 4, a(n) == 3; for n == 5, a(n) == 1.
For all n, a(n)+1 is a multiple of n^2.
For n odd  and n >= 3, a(n)-1 is a multiple of (n+1)^2.
For n even and n >= 0, a(n)+1 is a multiple of (n+1)^2.
For proofs of the above, see the Englander link. (End)

			a(2) = 1^2 - 2^1 = -1,
a(4) = 3^4 - 4^3 = 17.

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

T. D. Noe, Table of n, a(n) for n = 0..100
Mathew Englander, Notes on OEIS A007925
Sergio Silva, Teste Numerico, Item 3.
H. J. Smith, Contour Plot of z = x^y - y^x

Cf. A051442.

Cf. A166326, A099498, A141074, A174379, A123206, A045575, A082754.

A007925:=n->n^(n+1)-(n+1)^n: seq(A007925(n), n=0..25); # Wesley Ivan Hurt, Jan 10 2017

lst={};Do[AppendTo[lst, (n^(n+1)-((n+1)^n))], {n, 0, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
#^(#+1)-(#+1)^#&/@Range[0,20] (* Harvey P. Dale, Oct 22 2011 *)

A007925[n]:=n^(n+1)-(n+1)^n$ makelist(A007925[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n)=n^(n+1)-(n+1)^n \\ Charles R Greathouse IV, Feb 06 2017

Asymptotic expression for a(n) is a(n) ~ n^n * (n - e). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
From Mathew Englander, Jul 07 2020: (Start)
a(n) = A111454(n+4) - 1.
a(n) = A055651(n, n+1).
a(n) = A220417(n+1, n) for n >= 1.
a(n) = A007778(n) - A000169(n+1).
(End)
E.g.f.: LambertW(-x)/((1+LambertW(-x))*x)-LambertW(-x)/(1+LambertW(-x))^3. - Alois P. Heinz, Jul 04 2022

A174380 Smallest prime factors of numbers of the form (n-1)^n - n^(n-1) A007925.

Original entry on oeis.org

17, 3, 47, 162287, 23, 257, 2486784401, 3, 33479, 83695120256591, 5, 67, 7, 3, 443881, 100417, 859, 79, 10745792197529, 3, 92798617729, 67, 1627, 11, 12298336501553, 3, 19, 241, 167, 3506869732968391733353, 5, 3, 47, 5, 317

Offset: 4

Torbjorn Alm (talm(AT)tele2.se), Mar 17 2010

nonn

For n = 1 to 3 no prime factors in A007925(n).

A007925, A174379

Table[FactorInteger[(n-1)^n-n^(n-1)][[1,1]],{n,4,40}] (* Harvey P. Dale, May 25 2011 *)

cp's OEIS Frontend

A007925 a(n) = n^(n+1) - (n+1)^n.

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