A048861 a(n) = n^n - 1.

0, 3, 26, 255, 3124, 46655, 823542, 16777215, 387420488, 9999999999, 285311670610, 8916100448255, 302875106592252, 11112006825558015, 437893890380859374, 18446744073709551615, 827240261886336764176, 39346408075296537575423, 1978419655660313589123978

Offset: 1

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

• nonn
• easy

Comments

From Alexander Adamchuk, Jan 22 2007: (Start)

a(n) is divisible by (n-1).

Corresponding quotients are a(n)/(n-1) = {1,3,13,85,781,9331, ...} = A023037(n).

p divides a(p-1) for prime p.

p divides a((p-1)/2) for prime p = {3,11,17,19,41,43,59,67,73,83,89,97,...} = A033200 Primes congruent to {1, 3} mod 8; or, odd primes of form x^2+2*y^2.

p divides a((p-1)/3) for prime p = {61,67,73,103,151,193,271,307,367,...} = A014753 3 and -3 are both cubes (one implies other) mod these primes p=1 mod 6.

p divides a((p-1)/4) for prime p = {5,13,17,29,37,41,53,61,73,...} = A002144 Pythagorean primes: primes of form 4n+1.

p divides a((p-1)/5) for prime p = {31,191,251,271,601,641,761,1091,...}.

p divides a((p-1)/6) for prime p = {7,241,313,337,409,439,607,631,727,751,919,937,...}. (End)

For n > 1, a(n) is largest number that can be represented using n digits in the base-n number system. - Chinmaya Dash, Mar 31 2022

Examples

			For n=3, a(n) = 3^3 - 1 = 27 - 1 = 26. - _Michael B. Porter_, Nov 12 2017
		

References

• M. Le, Primes in the sequences n^n+1 and n^n-1, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 156-157.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..385
• F. Smarandache, Only Problems, Not Solutions!

Crossrefs

Cf. A000312, A048860, A023037, A033200, A014753, A002144.

Programs

Magma

[ n^n-1: n in [1..25]]; // Vincenzo Librandi, Dec 29 2010

Mathematica

Table[n^n - 1, {n, 1, 50}] (* G. C. Greubel, Nov 10 2017 *)

PARI

a(n)=n^n-1 \\ Charles R Greathouse IV, Feb 24 2012

Formula

E.g.f.: 1/(1+LambertW(-x)) - exp(x). - Vaclav Kotesovec, Dec 20 2014

Extensions

Extended (and corrected) by Patrick De Geest, Jul 15 1999