The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A343009 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A343009

a(n) = (n^(2n)-1)/(n^2-1) for n > 1, a(1) = 1.
(history; published version)

#29 by Michael De Vlieger at Sun Aug 28 21:11:28 EDT 2022

STATUS

proposed

approved

#28 by Jianing Song at Sun Aug 28 20:56:34 EDT 2022

STATUS

editing

proposed

#27 by Jianing Song at Sun Aug 28 20:56:27 EDT 2022

COMMENTS

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.

#26 by Jianing Song at Sun Aug 28 20:54:29 EDT 2022

COMMENTS

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, 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)

STATUS

approved

editing

#25 by Michael De Vlieger at Sun Aug 28 15:21:41 EDT 2022

STATUS

proposed

approved

#24 by Alois P. Heinz at Sun Aug 28 14:02:46 EDT 2022

STATUS

editing

proposed

#23 by Alois P. Heinz at Sun Aug 28 14:02:14 EDT 2022

EXTENSIONS

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

#22 by Alois P. Heinz at Sun Aug 28 13:59:30 EDT 2022

FORMULA

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

CROSSREFS

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

#21 by Alois P. Heinz at Sun Aug 28 13:55:53 EDT 2022

FORMULA

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

#20 by Alois P. Heinz at Sun Aug 28 13:55:15 EDT 2022

STATUS

proposed

editing