A089146 - OEIS

A089146

Greatest common divisor of n^2 - 4 and n^2 + 4.

4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4, 1, 8, 1, 4

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OFFSET

0,1

COMMENTS

Also decimal expansion of 4181/9999 = 0.418141814181...

Repeat [4,1,8,1], because for odd n, the GCD is 1; for n = 4k+2, GCD(16(k^2+k), 16(k^2+k)+8) = 8; for n = 4k, (16k^2-4,16k^2+4) can be divided by 4, but then GCD(4k^2-1,4k^2+1) = 1. - Georg Fischer, Jul 21 2022

LINKS

Table of n, a(n) for n=0..100.

FORMULA

Multiplicative with a(2) = 8, a(2^e) = 4 if e >= 2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005

Dirichlet g.f.: zeta(s)*(1+7/2^s-4^(1-s)). - Amiram Eldar, Dec 31 2022

Sum_{k=1..n} a(k) ~ 7*n/2. - Vaclav Kotesovec, Dec 31 2022

MATHEMATICA

a[n_] := GCD[n^2 - 4, n^2 + 4]; Array[a, 101, 0] (* Amiram Eldar, Dec 31 2022 *)

PROG

(PARI) g(n) = for(x=0, n, print1(gcd(x^2-4, x^2+4)", "))

CROSSREFS

Cf. A028347, A087475.