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A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes. 3
89391959, 89591959, 519512471, 519512473, 531324041, 561324041, 699023791, 699023891, 874481011, 874487011, 1862537503, 2232483271, 2232483871, 2608559351, 3127181789, 3157181789, 3928401949, 3928401989, 4070171669, 4070171969, 5225628323, 5309756339, 5525628323 (list; graph; refs; listen; history; text; internal format)



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.


Michael Kleber, Table of n, a(n) for n = 1..66


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]]]]


Cf. A158576, A050249, A158124.

Sequence in context: A234981 A054434 A164850 * A227654 A069318 A172573

Adjacent sequences: A253266 A253267 A253268 * A253270 A253271 A253272




Michael Kleber, May 01 2015



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Last modified December 1 07:34 EST 2022. Contains 358454 sequences. (Running on oeis4.)