A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

6, 5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1

Offset: 0

Author

Paul Barry, Jun 09 2006

Keywords

• easy
• nonn

Comments

The subdiagonal of A120113 is -a(n).

From Robert Israel, Dec 03 2024: (Start)

a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.

Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.

if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;

if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);

otherwise a(n) is odd. (End)

Conjectures from Davide Rotondo, Dec 02 2024: (Start)

Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.

If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000961, A099996, A120113.

Programs

GAP

List([0..75],n->Lcm(List([1..2*n+4],i->i))/Lcm(List([1..2*n+2],i->i))); # Muniru A Asiru, Mar 04 2019

Magma

A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
[A120114(n): n in [0..100]]; // G. C. Greubel, May 05 2023

Maple

f:= proc(n) local t,x,S;
   t:= 1;
   for x from 2*n+3 to 2*n+4 do
     S:= numtheory:-factorset(x);
     if nops(S) = 1 then t:= t*S[1] fi;
   od;
   t
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2024

Mathematica

Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* Harvey P. Dale, Dec 15 2017 *)

SageMath

def A120114(n):
    return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
[A120114(n) for n in range(101)] # G. C. Greubel, May 05 2023

Formula

a(n) = A099996(n+2)/A099996(n+1). - Michel Marcus, May 06 2023

Extensions

More terms from Harvey P. Dale, Dec 15 2017