A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes.

89391959, 89591959, 519512471, 519512473, 531324041, 561324041, 699023791, 699023891, 874481011, 874487011, 1862537503, 2232483271, 2232483871, 2608559351, 3127181789, 3157181789, 3928401949, 3928401989, 4070171669, 4070171969, 5225628323, 5309756339, 5525628323

Offset: 1

Author

Michael Kleber, May 01 2015

Keywords

• nonn
• base

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

Crossrefs

Cf. A158576, A050249, A158124.

Programs

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]]]]