A117812 -id:A117812

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A280797

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

+10
0

0, 0, 5, 182, 13107, 1627604, 310968905, 84777884106, 31274997412295, 15009463529699912, 9090909090909090909, 6783562448903313426110, 6115142092568526471803195, 6552380728090615475599972572, 8231779712749862388415318790417, 11984441202055255416780710220336914

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..15.

FORMULA

a(n) = A117812(n)/(n + 1).

EXAMPLE

a(0) = 0 because (0^0 - 1)*(0^0 + 1)/(0 + 1) = 0,

a(1) = 0 because (1^1 - 1)*(1^1 + 1)/(1 + 1) = 0,

a(2) = 5 because (2^2 - 1)*(2^2 + 1)/(2 + 1) = 5.

MATHEMATICA

Table[If[n == 0, 0, (n^n - 1) (n^n + 1)/(n + 1)], {n, 0, 15}] (* Michael De Vlieger, Jan 09 2017 *)

PROG

(Magma) [(n^(2*n)-1)/(n+1): n in [0..15]];

CROSSREFS

Cf. A000312, A081216, A117812, A179897.

KEYWORD

nonn,easy,changed

AUTHOR

Juri-Stepan Gerasimov, Jan 08 2017

EXTENSIONS

Offset changed to 0 by Georg Fischer, Jul 15 2024

STATUS

approved

A343009

a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.

+10
0

1, 5, 91, 4369, 406901, 62193781, 14129647351, 4467856773185, 1876182941212489, 1010101010101010101, 678356244890331342611, 555922008415320588345745, 546031727340884622966664381, 633213824057681722185793753109, 856031514432518244055765015738351

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: for n > 2, a(n) is a Fermat pseudoprime to base n.

If p is an odd prime, then a(p) is a Cipolla pseudoprime to base p.

Is a(m) a Fermat pseudoprime to base m for every composite m?

Amiram Eldar confirmed this up to m = 3800.

From Jianing Song, Aug 28 2022: (Start)

a(n) = Product_{d|(2n),d>2} Phi(d,n), where Phi(n,x) is the d-th cyclotomic polynomial. Note that Phi(n,x) > 1 for x >= 2 unless (n,x) = (1,2): suppose that n >= 3 and x >= 2, then Phi(n,x) = Product_{1<=j<=n,gcd(j,n)=1} (x - exp(2*j*Pi*i/n)) = Product_{1<=j<=n/2,gcd(j,n)=1} (x^2 - 2*cos(2*j*Pi/n)*x + 1) = Product_{1<=j<=n/2,gcd(j,n)=1} ((x - cos(2*j*Pi/n))^2 + (sin(2*j*Pi/n))^2) > 1 since x - cos(2*j*Pi/n) > 1. This shows that a(n) is composite for n > 2.

For n > 2, a(n) is a Fermat pseudoprime to base n, since n^(2*n) == 1 (mod a(n)) and 2*n divides a(n)-1 = n^2*(n^(2*n-2)-1)/(n^2-1): if n is even, then 2*n | n^2; if n is odd, then n | n^2 and 2 | n^2+1 = (n^4-1)/(n^2-1) | (n^(2*n-2)-1)/(n^2-1). (End)

LINKS

Table of n, a(n) for n=1..15.

FORMULA

a(n) = Sum_{k=0..n-1} n^(2*k). - Davide Rotondo, Aug 28 2022

From Alois P. Heinz, Aug 28 2022: (Start)

a(n) = A117812(n)/A005563(n-1) = A117812(n)/A132411(n-1) for n>=2.

Limit_{n -> 1} (n^(2*n)-1)/(n^2-1) = 1. (End).

EXAMPLE

a(10) = (10^20-1)/99 = 1010101010101010101.

MATHEMATICA

a[n_] := (n^(2*n)-1)/(n^2-1); a[1] = 1; Array[a, 15] (* Amiram Eldar, Apr 02 2021 *)

CROSSREFS

Cf. A005563, A023037, A117812, A124899, A132411, A210461.

KEYWORD

nonn

AUTHOR

Thomas Ordowski, Apr 02 2021

EXTENSIONS

More terms from Amiram Eldar, Apr 02 2021

a(1)=1 prepended and name adapted by Alois P. Heinz, Aug 28 2022

STATUS

approved

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