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

%I A253269 #25 Nov 06 2015 13:31:59
%S A253269 89391959,89591959,519512471,519512473,531324041,561324041,699023791,
%T A253269 699023891,874481011,874487011,1862537503,2232483271,2232483871,
%U A253269 2608559351,3127181789,3157181789,3928401949,3928401989,4070171669,4070171969,5225628323,5309756339,5525628323
%N A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes.
%C A253269 Each pair of twins here form a size-two connected component in the graph considered in A158576.
%C A253269 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.
%C A253269 This is analogous to a theorem about weakly prime numbers; see the Terence Tao paper referenced in A050249.
%H A253269 Michael Kleber, <a href="/A253269/b253269.txt">Table of n, a(n) for n = 1..66</a>
%t A253269 NeighborsAndSelf[n_] := Flatten[MapIndexed[Table[ n + (i - #)*10^(#2[[1]] - 1), {i, 0, 9}] &, Reverse[IntegerDigits[n, 10]]]]
%t A253269 PrimeNeighbors[n_] := Complement[Select[NeighborsAndSelf[n],PrimeQ],{n}]
%t A253269 WeaklyTwinPrime[p_] := (Length[#] == 1 && PrimeNeighbors[#[[1]]] == {p}) &[PrimeNeighbors[p]]
%t A253269 For[k = 0, k <= PrimePi[10^10], k++, If[WeaklyTwinPrime[Prime[k]], Print[Prime[k]]]]
%Y A253269 Cf. A158576, A050249, A158124.
%K A253269 nonn,base
%O A253269 1,1
%A A253269 _Michael Kleber_, May 01 2015