A023037 a(n) = n^0 + n^1 + ... + n^(n-1), or a(n) = (n^n-1)/(n-1) with a(0)=0; a(1)=1.

0, 1, 3, 13, 85, 781, 9331, 137257, 2396745, 48427561, 1111111111, 28531167061, 810554586205, 25239592216021, 854769755812155, 31278135027204241, 1229782938247303441, 51702516367896047761, 2314494592664502210319, 109912203092239643840221

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

For prime n, a(n) is conjectured to be the period of Bell numbers (mod n). See A054767. - T. D. Noe, Oct 12 2007

For prime n, a(n) is a multiple of the period of Bell numbers mod n (and conjectured to be exactly the period, as mentioned above). - Charles R Greathouse IV, Jul 31 2012

For n >= 1, a(n) is the number whose base n representation is a string of n ones. For example, 11111 in base 5 is a(5) = 781. - Melvin Peralta, May 23 2016

For n > 0, n^(a(n)-1) == 1 (mod a(n)), so for n > 1, a(n) is a prime or a Fermat pseudoprime to base n. - Thomas Ordowski, Mar 15 2021

Examples

			a(3) = 3^0 + 3^1 + 3^2 = 1+3+9 = 13.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..387 (first 101 terms from T. D. Noe)
• Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
• W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16.

Crossrefs

Cf. A001039, A054767, A088790 (n such that a(n) is prime), A125118.

Programs

Maple

A023037:=n->add(n^i, i=0..n-1): seq(A023037(n), n=0..25); # Wesley Ivan Hurt, May 28 2016

Mathematica

Join[{0,1},Table[(n^n-1)/(n-1),{n,2,20}]] (* Harvey P. Dale, Aug 01 2014 *)

PARI

a(n) = if(n==1, 1, (n^n-1)/(n-1)); \\ Altug Alkan, Oct 04 2017

Python

def A023037(n): return (n**n-1)//(n-1) if n>1 else n # Chai Wah Wu, Sep 28 2023

Sage

[lucas_number1(n,n+1,n) for n in range(0, 19)] # Zerinvary Lajos, May 16 2009

Formula

a(n) = A125118(n,n-1) for n>1. - Reinhard Zumkeller, Nov 21 2006

a(n) = [x^n] x/((1 - x)*(1 - n*x)). - Ilya Gutkovskiy, Oct 04 2017

Extensions

Entry improved by Tobias Nipkow (nipkow(AT)in.tum.de).