login
Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.
3

%I #31 Apr 21 2022 14:04:57

%S 2,3,4,5,8,11,59,1319,1619,4259,5099,6659,6779,11699,12539,21059,

%T 66359,83219,88259,107099,110879,114659,127679,130199,140759,141959,

%U 144539,148199,149519,157559,161339,163859,175079,186479,204599,230939,249539,267959,273899,312839

%N Numbers m such that m-2, m-1, m+1, m+2 cannot all be represented in the form x*y + x + y for values x, y with x >= y > 1.

%C Indices of terms surrounded by pairs of zeros in A255361.

%C Conjectures:

%C 1. A255361(a(n)) > 0 for n > 4.

%C 2. All terms > 8 are primes.

%C 3. All terms > 8 are terms of these supersequences: A118072, A171667, A176821, A181602, A181669.

%C From _Lamine Ngom_, Feb 12 2022: (Start)

%C For n > 4, a(n) is not a term of A254636. This means that a(n)-2, a(n)-1, a(n)+1 and a(n)+2 are adjacent terms in A254636.

%C Number of terms < 10^k: 5, 7, 7, 13, 19, 96, 441, 2552, ...

%C Conjecture 2 would follow if we establish the equivalence "t is in sequence" <=> "t is a term of b(n): lesser of twin primes pair p and q such that (p - 1)/2 and (q + 1)/2 are also a pair of twin primes (A077800)".

%C It appears that b(n) = a(n) for n > 5. Verified for all terms < 10^9. (End)

%F a(n) = A158870(n-5) - 2, n > 5 (conjectured). - _Lamine Ngom_, Feb 12 2022

%e 9, 10, 12, 13 cannot be represented as x*y + x + y, where x >= y > 1. Therefore 11 is in the sequence.

%Y Cf. A255361, A254636.

%Y Cf. A118072, A171667, A176821, A181602, A181669.

%Y Cf. A001097, A001359, A006512, A077800, A158870.

%K nonn

%O 1,1

%A _Alex Ratushnyak_, Mar 31 2015