A244441 a(n) is the smallest prime p such that p has at least one digit greater than 1 and all the n numbers ds(p), ds(ds(p)), ..., ds(ds(...(ds(p))...)) are primes. The function ds is defined in the comment lines.

2, 31, 2621, 9941, 5599921, 5599921, 5219088341

Offset

1,1

Comments

ds(m) is the number that is obtained from m by replacing each positive digit i to sigma(i) and replacing zero by zero itself.

Example: ds(19)=1.13=113, ds(1028)=1.0.3.15=10315.

1. If all digits of m are less than 2 then ds(m)=m. So for

primes p with digits less than 2 (terms of the sequence A020449) p=ds(p)=ds(ds(p))=ds(ds(ds(p)))= ... .

2. For n>1, a(n) is of the form 10k+1.

3. If o, i, s and t are respectively number of zeros, number of ones, number of digits greater than 1 and number of composite digits greater than 4 in decimal expansion of m also o', i', s' and t' are the same for ds(m) then o'=o, i'=i+t and s'=s.

Example: m=1021041629839

ds(m)=1.0.3.1.0.7.1.12.3.13.15.4.13=10310711231315413

=> {o, i, s, t}={2, 3, 8, 4} and {o', i', s'}={o, i+t, s}= {2, 7, 8}.

Links

Table of n, a(n) for n=1..7.

Example

a(3)=2621 because the three numbers:
1. ds(2621)=3.12.3.1=31231
2. ds(ds(2621))=4.1.3.41=41341
3. ds(ds(ds(2621)))=7.1.4.71=71471
are all primes and 2621 is the smallest prime with this property.

Crossrefs

Cf. A000040, A020449, A244442.

Sequence in context: A197320 A239332 A350940 * A004072 A113030 A247873

Adjacent sequences: A244438 A244439 A244440 * A244442 A244443 A244444

Keyword

nonn,base,hard,more

Author

Jahangeer Kholdi and Farideh Firoozbakht, Jul 13 2014

Status

approved