

A166011


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


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
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OFFSET

1,1


COMMENTS

From Altug Alkan, Apr 22 2016: (Start)
For n > 1, a(n) is (p3)*(p+3)/2 where p is the nth prime. The reason is that the greatest common divisor of p3 and p+3 is always 2 where p is the nth prime and n > 2.
Proof: Let us assume that q is the greatest common divisor of p3 and p+3. Because of the fact that any divisor of a and b must divide ab, 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 p3 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 p3 and p+3 is always 2 where p is the nth prime and n > 2, then the least common multiple of p3 and p+3 must be (p3)*(p+3)/2 where p is the nth prime and n > 2 because of the general identity lcm(a, b) * gcd(a, b) = a*b. Note that for p = 3, (p3)*(p+3)/t always is equal to 0 for any nonzero integer t, so it can be said that a(n) is (p3)*(p+3)/2 where p is the nth prime and n > 1. (End)


LINKS

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


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[n3, 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 *)


PROG

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


CROSSREFS

Cf. A066830, A084921, A166010.
Sequence in context: A240358 A200422 A141431 * A344144 A266222 A266439
Adjacent sequences: A166008 A166009 A166010 * A166012 A166013 A166014


KEYWORD

nonn,easy


AUTHOR

Vladimir Joseph Stephan Orlovsky, Oct 04 2009


STATUS

approved



