A166011 Least common multiple of prime(n)-3 and prime(n)+3.

5, 0, 8, 20, 56, 80, 140, 176, 260, 416, 476, 680, 836, 920, 1100, 1400, 1736, 1856, 2240, 2516, 2660, 3116, 3440, 3956, 4700, 5096, 5300, 5720, 5936, 6380, 8060, 8576, 9380, 9656, 11096, 11396, 12320, 13280, 13940, 14960, 16016, 16376, 18236, 18620

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Oct 04 2009

Keywords

• nonn
• easy

Comments

From Altug Alkan, Apr 22 2016: (Start)

For n > 1, a(n) is (p-3)*(p+3)/2 where p is the n-th prime. The reason is that the greatest common divisor of p-3 and p+3 is always 2 where p is the n-th prime and n > 2.

Proof: Let us assume that q is the greatest common divisor of p-3 and p+3. Because of the fact that any divisor of a and b must divide a-b, we know that q must divide 6. Note that q cannot be a multiple of 3 because p is prime, that is, q must be 1 or 2. Since we know that p-3 and p+3 are always even numbers for odd prime p, q must be 2 because we define it as the greatest common divisor.

If the greatest common divisor of p-3 and p+3 is always 2 where p is the n-th prime and n > 2, then the least common multiple of p-3 and p+3 must be (p-3)*(p+3)/2 where p is the n-th prime and n > 2 because of the general identity lcm(a, b) * gcd(a, b) = a*b. Note that for p = 3, (p-3)*(p+3)/t always is equal to 0 for any nonzero integer t, so it can be said that a(n) is (p-3)*(p+3)/2 where p is the n-th prime and n > 1. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A066830, A084921, A166010.

Programs

Maple

A166011:=n->lcm(ithprime(n)+3,ithprime(n)-3): seq(A166011(n), n=1..100); # Wesley Ivan Hurt, Apr 22 2016

Mathematica

f[n_]:=LCM[n-3,n+3]; lst={};Do[p=Prime[n];AppendTo[lst,f[p]],{n,5!}]; lst
LCM[#+3,#-3]&/@Prime[Range[50]] (* Harvey P. Dale, Aug 09 2015 *)

PARI

a(n) = lcm(prime(n)-3, prime(n)+3); \\ Michel Marcus, Apr 22 2016