OFFSET
1,1
COMMENTS
Each pair of twins here form a size-two connected component in the graph considered in A158576.
A naive heuristic argument based on the density of primes claims that this sequence should be infinite, and in fact that a positive proportion of all primes should have this property. A prime p has 9*log_10(p) neighbors, each prime with "probability" 1/log(p), and with all the other 2*9*log_10(p) neighbors being composite with "probability" (1-1/log(p))^(2*9*log_10(p)). For a large prime p, this goes to the limit 9/(exp(18/log(10))*log(10)), or about 0.16%. The fact that base-10 primes need to end with digit 1/3/7/9 will change the value of this probability, but won't change the fact that it is nonzero.
This is analogous to a theorem about weakly prime numbers; see the Terence Tao paper referenced in A050249.
LINKS
Michael Kleber, Table of n, a(n) for n = 1..66
MATHEMATICA
NeighborsAndSelf[n_] := Flatten[MapIndexed[Table[ n + (i - #)*10^(#2[[1]] - 1), {i, 0, 9}] &, Reverse[IntegerDigits[n, 10]]]]
PrimeNeighbors[n_] := Complement[Select[NeighborsAndSelf[n], PrimeQ], {n}]
WeaklyTwinPrime[p_] := (Length[#] == 1 && PrimeNeighbors[#[[1]]] == {p}) &[PrimeNeighbors[p]]
For[k = 0, k <= PrimePi[10^10], k++, If[WeaklyTwinPrime[Prime[k]], Print[Prime[k]]]]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Michael Kleber, May 01 2015
STATUS
approved