A068955 -id:A068955 - 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

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

A127854 Largest number k such that k^2 divides A007781(6n+1).

Original entry on oeis.org

19, 61, 127, 217, 331, 469, 631, 817, 1027, 1261, 1519, 1801, 2107, 2437, 2791, 3169, 3571, 3997, 4447, 4921, 5419, 5941, 6487, 7057, 7651, 8269, 8911, 9577, 10267, 10981, 11719, 12481, 13267, 14077, 14911, 15769, 16651, 17557, 18487, 19441

Offset: 1

Alexander Adamchuk, Apr 05 2007

nonn

A007781(n) = (n+1)^(n+1) - n^n. A007781(6n+1) is not squarefree for n > 0. a(n) is the largest square divisor of A007781(6n+1). All terms belong to A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence for hexagonal lattice). It appears that a(n) = A003215(2n) = 6n(2n+1)+1. A007781(6n+1)/A003215(2n)^2 = ((6n+2)^(6n+2)-(6n+1)^(6n+1))/(6n(2n+1)+1)^2 = {44193, 2904899682603, 6378521938392937343349, 128847538453506016002947264859159, 13183819636551142123977274666051092130410345, ...}. Prime terms of a(n) belong to A002407. Factorizations of the terms of a(n) are {19, 61, 127, 7*31, 331, 7*67, 631, 19*43, 13*79, 13*97, 7*7*31, 1801, 7*7*43, 2437, 2791, 3169, 3571, 7*571, 4447, 7*19*37, 5419, 13*457, 13*499, 7067, 7*1093, 8269, 7*19*67, 61*157, 10267, 79*139, ...}. All prime factors of a(n) are of the form 6k+1.

Cf. A007781 = (n+1)^(n+1) - n^n. Cf. A000312, A068955, A003215, A002407.

Conjecture: a(n) = 12n^2 + 6n + 1.
Conjecture: a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); g.f.: x*(19 + 4*x + x^2)/(1-x)^3. - Colin Barker, Mar 16 2012
These conjectures are false. For n=74, 12*n^2 + 6*n + 1 = 66157 but A007781(6*74+1) is divisible by 5491031^2. - Robert Israel, Nov 19 2017

a(24) corrected by T. D. Noe, Mar 14 2008

A162591 Primes in A007781.

Original entry on oeis.org

3, 23, 229, 776887, 275311670611, 808793517812627212561, 47962816398523117606189726043968411848519304708598059350620557763277694737755820158580941773369740112983781265183299561695077810144494290292906506606685128216915382107158604900927276535058149770652889252352435564631

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

nonn

See A072164 for a condensed representation of the same information.

			3^3-2^2=27-4=23 is prime and enters the list.

Cf. A068955.

f[n_]:=n^n-(n-1)^(n-1); lst={};Do[If[PrimeQ[f[n]],AppendTo[lst,f[n]]], {n,2,5!}];lst

Definition simplified, reference to A072164 and A068955 added by R. J. Mathar, Aug 11 2009

cp's OEIS Frontend

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

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

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A127854 Largest number k such that k^2 divides A007781(6n+1).

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A162591 Primes in A007781.

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