A205668 Prime numbers that cannot be expressed as the sum of two lesser primes of twin prime pairs + 1 or two greater primes of twin prime pairs - 1.

2, 3, 5, 97, 401, 787, 907, 1117

Offset: 1

Author

Manuel Valdivia, Jan 30 2012

Keywords

• nonn

Comments

The occurrence of a pair of twin primes in the sequence would be a counterexample to the conjecture in A134143.

There are probably no more terms. As in Goldbach's conjecture, the number of summands increases rapidly. - Charles R Greathouse IV, Jan 31 2012

Examples

			97 is here because neither 96 or 98 is the sum of two primes from the set {2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73}, which are the twin primes less than 100. - _T. D. Noe_, Feb 12 2012
		

Crossrefs

Cf. A000040, A001359, A006512.

Programs

Mathematica

k=Insert[Select[Prime[Range[2,10^4]], PrimeQ[#-2]||PrimeQ[#+2]&], 5, 3]; u=Length@k/2; Complement[Prime[Range[4,10^4]], Select[Flatten[Join[Table[k[[2n-1]] + k[[2m-1]] + 1,{n,u}, {m,n}], Table[k[[2n]] + k[[2m]] - 1,{n,u}, {m,n}]]], PrimeQ]]

PARI

lower=List();p=2;forprime(q=3,1e8,if(q-p==2,listput(lower,p));p=q)
isk(n)=for(i=1,#lower,if(setsearch(lower,n-lower[i]),return(lower[i]));if(2*lower[i]>n,return(0)));error("ran out")
is(n)=!isk(n-1)&&!isk(n-3) \\ Charles R Greathouse IV, Jan 31 2012

Formula

A000040(n) != A001359(j) + A001359(k) + 1 and A000040(n) != A006512(j) + A006512(k) - 1, with n>3 and j<=k.