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

1,1

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

Links

Table of n, a(n) for n=1..8.

Formula

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

Example

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

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]]

Prog

(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

Crossrefs

Cf. A000040, A001359, A006512.

Sequence in context: A065406 A375781 A111331 * A118505 A067799 A321362

Adjacent sequences: A205665 A205666 A205667 * A205669 A205670 A205671

Keyword

nonn

Author

Manuel Valdivia, Jan 30 2012

Status

approved