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

%I A120114 #38 Dec 16 2024 14:57:24
%S A120114 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,
%T A120114 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,
%U A120114 1,113,1,1,1,11,1,5,254,1,131,1,1,137,139,1,1,1,1
%N A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).
%C A120114 The subdiagonal of A120113 is -a(n).
%C A120114 From _Robert Israel_, Dec 03 2024: (Start)
%C A120114 a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.
%C A120114 Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.
%C A120114       if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;
%C A120114       if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);
%C A120114       otherwise a(n) is odd. (End)
%C A120114 Conjectures from _Davide Rotondo_, Dec 02 2024: (Start)
%C A120114 Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.
%C A120114 If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)
%H A120114 Muniru A Asiru, <a href="/A120114/b120114.txt">Table of n, a(n) for n = 0..5000</a>
%F A120114 a(n) = A099996(n+2)/A099996(n+1). - _Michel Marcus_, May 06 2023
%p A120114 f:= proc(n) local t,x,S;
%p A120114    t:= 1;
%p A120114    for x from 2*n+3 to 2*n+4 do
%p A120114      S:= numtheory:-factorset(x);
%p A120114      if nops(S) = 1 then t:= t*S[1] fi;
%p A120114    od;
%p A120114    t
%p A120114 end proc:
%p A120114 map(f, [$0..100]); # _Robert Israel_, Dec 03 2024
%t A120114 Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* _Harvey P. Dale_, Dec 15 2017 *)
%o A120114 (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
%o A120114 (Magma)
%o A120114 A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
%o A120114 [A120114(n): n in [0..100]]; // _G. C. Greubel_, May 05 2023
%o A120114 (SageMath)
%o A120114 def A120114(n):
%o A120114     return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
%o A120114 [A120114(n) for n in range(101)] # _G. C. Greubel_, May 05 2023
%Y A120114 Cf. A000961, A099996, A120113.
%K A120114 easy,nonn
%O A120114 0,1
%A A120114 _Paul Barry_, Jun 09 2006
%E A120114 More terms from _Harvey P. Dale_, Dec 15 2017