A124923 a(n) = n^(n-1) + 1.

2, 3, 10, 65, 626, 7777, 117650, 2097153, 43046722, 1000000001, 25937424602, 743008370689, 23298085122482, 793714773254145, 29192926025390626, 1152921504606846977, 48661191875666868482, 2185911559738696531969

Offset: 1

Author

Alexander Adamchuk, Nov 12 2006

Keywords

• nonn
• easy

Comments

Prime p divides a(p-1). n divides a(n-1) for all prime n and all odd composite n.

p divides a((p+1)/2) for prime p = {3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, ...} = A003629 (Primes congruent to {3,5} mod 8).

p divides a((p+3)/4) for prime p = {13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457, 541, 709, 733, 757, 829, ...} = A107141 (Primes of the form 4x^2+9y^2).

p divides a((p+5)/6) for prime p = {43, 61, 79, 109, 151, 163, 181, 193, 313, 337, 433, 523, 577, 631, 643, 673, 787, 829, 907, 991, ...}.

p divides a((p+7)/8) for prime p = {113, 137, 569, 641, 673, 1129, 1289, 1297, 1481, 1801, ...}.

p divides a((3p-1)/2) for prime p = {5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, ...} = A003628 (Primes congruent to {5, 7} mod 8).

p^2 divides a((3p-1)/2) for prime p = {5, 13, 173, 5501, ...} = A124924.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000169, A003628, A003629, A107141, A124924.

Programs

GAP

List([1..20], n-> n^(n-1) + 1); # G. C. Greubel, Nov 19 2019

Magma

[n^(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Aug 14 2012

Maple

seq(n^(n-1) + 1, n=1..20); # G. C. Greubel, Nov 19 2019

Mathematica

Table[n^(n-1)+1, {n,20}]

PARI

vector(20, n, n^(n-1) + 1) \\ G. C. Greubel, Nov 19 2019

Sage

[n^(n-1) + 1 for n in (1..20)] # G. C. Greubel, Nov 19 2019

Formula

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

a(n) = A000169(n) + 1.

E.g.f.: -1 + exp(x) - W(-x), where W(x) is the Lambert w-function. - G. C. Greubel, Nov 19 2019