login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)
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

Cf. A158576, A050249, A158124.

Sequence in context: A234981 A054434 A164850 * A227654 A069318 A172573

Adjacent sequences: A253266 A253267 A253268 * A253270 A253271 A253272

KEYWORD

nonn,base

AUTHOR

Michael Kleber, May 01 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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