# [seqfan] 10-digit accepting integers

Eric Angelini Eric.Angelini at kntv.be
Mon Oct 27 17:12:51 CET 2014

```Hello Seqfans,
There is an infinity of such 10-digit accepting numbers!

Take 43021123 for instance (where 2 adjacent digits sum to 7,3,2,3,2,3,5 = primes)

43021123 + 0 = 430211230 -- Two adjacent digits sums = 7,3,2,3,2,3,5,3
43021123 + 1 = 143021123 -- Two adjacent digits sums = 5,7,3,2,3,2,3,5
43021123 + 2 = 430211232 -- Two adjacent digits sums = 7,3,2,3,2,3,5,5
43021123 + 3 = 343021123 -- Two adjacent digits sums = 7,7,3,2,3,2,3,5
43021123 + 4 = 430211234 -- Two adjacent digits sums = 7,3,2,3,2,3,5,7
43021123 + 5 = 430521123 -- Two adjacent digits sums = 7,3,5,7,3,2,3,5
43021123 + 6 = 430216123 -- Two adjacent digits sums = 7,3,2,3,7,7,3,5
43021123 + 7 = 743021123 -- Two adjacent digits sums = 11,7,3,2,3,2,3,5
43021123 + 8 = 430211238 -- Two adjacent digits sums = 7,3,2,3,2,3,5,11
43021123 + 9 = 943021123 -- Two adjacent digits sums = 13,7,3,2,3,2,3,5

43021123 produces immediately a family of 10-digit accepting integers
by simply expanding the "11" pattern into "111", "1111", 111...111":

43021123, 430211123, 4302111123, 4302111...11123

The question of the smallest such 10-digit accepting integer is still open.
Best,
É.

-----Message d'origine-----
De : SeqFan [mailto:seqfan-bounces at list.seqfan.eu] De la part de Eric Angelini
Envoyé : lundi 27 octobre 2014 12:53
À : Sequence Fanatics Discussion list
Objet : [seqfan] 9-digit accepting integers

Hello SeqFans,

21123 has the property that every pair of adjacent digits sum to a prime number

(thus belonging to https://oeis.org/A182175).

We want to transform this integer into another integer having the same property

by inserting single digits somewhere:

21123 + 0 = 211230

21123 + 1 = 121123

21123 + 2 = 211232

21123 + 3 = 321123

21123 + 4 = 211234

21123 + 5 = 521123

21123 + 6 = 216123

21123 + 7 = impossible

21123 + 8 = 211238

21123 + 9 = 921123

I think no integer will ever accept the 10 digits from 0 to 9.

Is 21123 the smallest accepting 9 digits? What are the others

(like 41123, accepting 7 now, but not 5)?

Best,

É.

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