A068956 -id:A068956 - 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

A068955 Greatest prime factor of n^n - (n-1)^(n-1).

Original entry on oeis.org

3, 23, 229, 151, 431, 776887, 14731, 109, 80317, 275311670611, 19395030961, 10423708597, 968299894201, 19428121, 165218809021364149, 808793517812627212561, 3979203955386313, 588489604729898953429, 2126173979464312447783, 5043293621028391, 90772326303985278570534379

Offset: 2

Reinhard Zumkeller, Mar 11 2002

nonn

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

Daniel Suteu and Amiram Eldar, Table of n, a(n) for n = 2..86 (terms 2..62 from Daniel Suteu)

Cf. A006530, A007781, A068954, A068956.

a:= n-> max(map(i-> i[1], ifactors(n^n-(n-1)^(n-1))[2])):
seq(a(n), n=2..23);  # Alois P. Heinz, Mar 10 2019

a[n_] := FactorInteger[n^n - (n-1)^(n-1)][[-1, 1]]; Array[a, 20, 2] (* Amiram Eldar, Feb 06 2020 *)

a(n) = vecmax(factor(n^n-(n-1)^(n-1))[,1]); \\ Daniel Suteu, Mar 10 2019

a(n) = A006530(A007781(n-1)).

a(18)-a(22) from Daniel Starodubtsev, Mar 10 2019

A068954 Smallest prime factor of n^n-(n-1)^(n-1).

Original entry on oeis.org

3, 23, 229, 19, 101, 776887, 3, 7, 29, 275311670611, 5, 28201, 3, 52489, 109, 808793517812627212561, 9680119, 5, 3, 1137694897331, 3697, 29, 6361, 10667, 3, 23, 17787551, 41393681953973, 7, 4211, 3, 461, 83, 19, 31, 983, 3, 5, 89, 2251, 250460976091, 109, 3, 29

Offset: 2

Reinhard Zumkeller, Mar 11 2002

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 2..107

Cf. A020639, A007781, A068955, A068956.

a(n) = A020639(A007781(n)).

a(20)-a(40) from Daniel Starodubtsev, Mar 10 2019

a(41)-a(45) from Chai Wah Wu, Jul 15 2019

A068957 Number of prime divisors of n^n - (n-1)^(n-1), counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 3, 2, 4, 3, 2, 1, 2, 3, 4, 2, 2, 3, 3, 4, 9, 4, 2, 2, 5, 4, 6, 3, 6, 4, 4, 2, 6, 7, 6, 4, 3, 4, 8, 6, 2, 7, 4, 7, 12, 6, 4, 5, 5, 7, 9, 5, 5, 6, 2, 5, 10, 4, 6, 5, 5, 3, 9, 4, 4, 2, 3, 4, 9, 4, 6, 4, 5, 7, 9, 13, 8, 4, 2, 5, 7

Offset: 2

Reinhard Zumkeller, Mar 11 2002

nonn

			A007781(14) = 10809131718965763 = 3 * 61^2 * 968299894201, therefore a(14) = 4.

Cf. A001222, A007781, A068956.

Table[ Apply[ Plus, Transpose[ FactorInteger[n^n - (n - 1)^(n - 1)]] [[ -1]]], {n, 2, 52}]

a(n) = A001222(A007781(n)).

Edited and extended by Robert G. Wilson v, Mar 15 2002

a(53)-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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A068955 Greatest prime factor of n^n - (n-1)^(n-1).

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Maple

Mathematica

PARI

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A068954 Smallest prime factor of n^n-(n-1)^(n-1).

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Author

Keywords

Examples

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Crossrefs

Formula

Extensions

A068957 Number of prime divisors of n^n - (n-1)^(n-1), counted with multiplicity.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions