

A173426


a(n) is obtained by starting with 1, sequentially concatenating all decimal numbers up to n, and then, starting from (n1), sequentially concatenating all decimal numbers down to 1.


31



1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345678910987654321, 123456789101110987654321, 1234567891011121110987654321, 12345678910111213121110987654321, 123456789101112131413121110987654321
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OFFSET

1,2


COMMENTS

The first prime in this sequence is the 20digit number a(10) = 12345678910987654321. On Jul 20 2015, Shyam Sunder Gupta reported on the Number Theory Mailing List that he has found what is probably the second prime in the sequence. This is the 2446th term, namely the 17350digit probable prime 1234567..244524462445..7654321.  N. J. A. Sloane, Jul 29 2015  Aug 03 2015
There are no other (PR)prime members in this sequence for n<60000.  Serge Batalov, Jul 29 2015
David Broadhurst gives heuristic arguments which suggest that this sequence contains infinitely many primes.
See A075023 and A075024 for the smallest and largest prime factor of the terms.  M. F. Hasler, Jul 29 2015
Using summation in decimal length clades, one can obtain analytical expressions for the sequence:
a(n) = A002275(n)^2, for 1 <= n < 10;
a(n) = (120999998998*10^(4*n28)  2*10^(2*n9) + 8790000000121)/99^2, for 10 <= n < 10^2;
a(n) = (120999998998*10^(6*n227)  (1099022*10^(6*n406) + 242*10^(3*n108)  1087789*10^191)/111^2 + 8790000000121)/99^2, for 10^2 <= n < 10^3; etc.  Serge Batalov, Jul 29 2015


REFERENCES

D. Broadhurst, Primes from concatenation: results and heuristics, Number Theory List, Aug 01 2015 and later postings.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..150
Shyam Sunder Gupta, Puzzle 794, Prime Puzzles Web Site.
S. S. Gupta, A new 17350 digit Symmetric Prime, NmbrThry List, July 20, 2015
FactorDB, (121*10^(4*n19)  1002*10^(4*n28)  2*10^(2*n9) + 879*10^10 + 121)/99^2.


FORMULA

a(n) = concatenate(1,2,3,...,n2,n1,n,n1,n2,...,3,2,1).


MATHEMATICA

Table[FromDigits[Flatten[IntegerDigits/@Join[Range[n], Reverse[Range[ n1]]]]], {n, 15}] (* Harvey P. Dale, Sep 02 2015 *)


PROG

(PARI) A173426(n)=eval(concat(vector(n*21, k, if(k<n, Str(k), n*2k)))) \\ M. F. Hasler, Jul 29 2015


CROSSREFS

This sequence and A002477 (Wonderful Demlo numbers) agree up to the 9th term.
Cf. A002275, A007908, A075023, A075024.
Sequence in context: A057139 A321687 A002477 * A261570 A068117 A080162
Adjacent sequences: A173423 A173424 A173425 * A173427 A173428 A173429


KEYWORD

nonn,base


AUTHOR

Umut Uludag, Feb 18 2010


EXTENSIONS

More terms from and minor edits by M. F. Hasler, Jul 29 2015


STATUS

approved



