A068957 -id:A068957 - cp's OEIS frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

1, 3, 23, 229, 2869, 43531, 776887, 15953673, 370643273, 9612579511, 275311670611, 8630788777645, 293959006143997, 10809131718965763, 426781883555301359, 18008850183328692241, 808793517812627212561

Offset: 0

Peter McCormack (peter.mccormack(AT)its.csiro.au)

(12n^2 + 6n + 1)^2 divides a(6n+1), where (12n^2 + 6n + 1) = (2n+1)^3 - (2n)^3 = A127854(n) = A003215(2n) are the hex (or centered hexagonal) numbers. The prime numbers of the form 12n^2 + 6n + 1 belong to A002407. - Alexander Adamchuk, Apr 09 2007

			a(14) = 10809131718965763 = 3 * 61^2 * 968299894201.

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see equation (6.7).

Doug Bell, Table of n, a(n) for n = 0..100
Andrew Cusumano, Problem H-656, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; A Sequence Tending To e, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
Ronald K. Hoeflin, Mega Test. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Power Difference Prime.

Cf. A000312, A068146, A068954, A068955, A068956, A068957.

Cf. A002407, A003215, A127854.

[1] cat [(n+1)^(n+1)-n^n: n in [1..20]]; // Vincenzo Librandi, Aug 19 2015

seq( `if`(n=0,1,(n+1)^(n+1) -n^n), n=0..20); # G. C. Greubel, Mar 05 2020

Join[{1},Table[(n+1)^(n+1)-n^n,{n,20}]]  (* Harvey P. Dale, Feb 09 2011 *)
Differences[Table[n^n,{n,0,20}]] (* Charles R Greathouse IV, Feb 09 2011 *)

first(m)=vector(m,i,i--;(i+1)^(i+1) - i^i) /* Anders Hellström, Aug 18 2015 */

[1]+[(n+1)^(n+1) -n^n for n in (1..20)] # G. C. Greubel, Mar 05 2020

a(n) = A000312(n+1) - A000312(n) for n>0, a(0) = 1.
a(n) = abs(discriminant(x^(n+1)-x+1)).
E.g.f.: W(-x)/(1+W(-x)) - W(-x)/((1+W(-x))^3*x) where W is the Lambert W function. - Robert Israel, Aug 19 2015
Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - Amiram Eldar, Jan 03 2022

A068956 Number of distinct prime factors of n^n - (n-1)^(n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3, 4, 7, 4, 2, 2, 5, 4, 5, 3, 6, 4, 4, 2, 4, 7, 6, 4, 3, 4, 7, 6, 2, 7, 4, 7, 10, 6, 4, 5, 5, 7, 7, 5, 5, 6, 2, 5, 8, 4, 6, 5, 5, 3, 5, 4, 4, 2, 3, 4, 8, 4, 6, 4, 5, 7, 5, 13, 8, 4, 2, 5, 6

Offset: 2

Reinhard Zumkeller, Mar 11 2002

nonn

			A007781(13) = 10809131718965763 = 3 * 61^2 * 968299894201, therefore a(14) = 3.

Cf. A001221, A007781, A068957.

Table[ Length[ FactorInteger[n^n - (n - 1)^(n - 1)]], {n, 2, 52}] (* Reinhard Zumkeller *)
PrimeNu[#]&/@(#[[2]]-#[[1]]&/@Partition[Table[n^n,{n,52}],2,1]) (* Harvey P. Dale, Aug 24 2014 *)

a(n) = A001221(A007781(n-1)).

Edited and extended by Robert G. Wilson v, Mar 15 2002
a(53)-a(54) from Alois P. Heinz, May 12 2017
a(55)-a(86) from Amiram Eldar, Feb 06 2020

cp's OEIS Frontend

A007781 a(n) = (n+1)^(n+1) - n^n for n>0, a(0) = 1.

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