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%I #82 Feb 19 2023 16:14:31
%S 1,121,12321,1234321,123454321,12345654321,1234567654321,
%T 123456787654321,12345678987654321,12345678910987654321,
%U 123456789101110987654321,1234567891011121110987654321,12345678910111213121110987654321,123456789101112131413121110987654321
%N a(n) is obtained by starting with 1, sequentially concatenating all decimal numbers up to n, and then, starting from n-1, sequentially concatenating all decimal numbers down to 1.
%C The first prime in this sequence is the 20-digit 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 17350-digit probable prime 1234567..244524462445..7654321. See A359148. - _N. J. A. Sloane_, Jul 29 2015 - Aug 03 2015
%C There are no other (PR)prime members in this sequence for n<60000. - _Serge Batalov_, Jul 29 2015
%C _David Broadhurst_ gives heuristic arguments which suggest that this sequence contains infinitely many primes.
%C See A075023 and A075024 for the smallest and largest prime factor of the terms. - _M. F. Hasler_, Jul 29 2015
%C Using summation in decimal length clades, one can obtain analytical expressions for the sequence:
%C a(n) = A002275(n)^2, for 1 <= n < 10;
%C a(n) = (120999998998*10^(4*n-28) - 2*10^(2*n-9) + 8790000000121)/99^2, for 10 <= n < 10^2;
%C a(n) = (120999998998*10^(6*n-227) - (1099022*10^(6*n-406) + 242*10^(3*n-108) - 1087789*10^191)/111^2 + 8790000000121)/99^2, for 10^2 <= n < 10^3; etc. - _Serge Batalov_, Jul 29 2015
%C Curiously, 1234567891010987654321 is also a prime (see A259937). - _N. J. A. Sloane_, Nov 30 2021
%D D. Broadhurst, Primes from concatenation: results and heuristics, Number Theory List, Aug 01 2015 and later postings.
%H G. C. Greubel, <a href="/A173426/b173426.txt">Table of n, a(n) for n = 1..150</a>
%H FactorDB, <a href="http://factordb.com/index.php?query=%28121*10%5E%284*n-19%29+-+1002*10%5E%284*n-28%29+-+2*10%5E%282*n-9%29+%2B+879*10%5E10+%2B+121%29%2F99%5E2&perpage=50">(121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2</a>.
%H Shyam Sunder Gupta, <a href="http://www.primepuzzles.net/puzzles/puzz_794.htm">Puzzle 794. Prime Generalized Palindromes</a>, The Prime Puzzles and Problems Connection.
%H S. S. Gupta, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;46e44b27.1507">A new 17350 digit Symmetric Prime</a>, NmbrThry List, July 20, 2015.
%H Brady Haran and N. J. A. Sloane, <a href="https://youtu.be/vKlVNFOHJ9I">The Most Wanted Prime Number</a>, Numberphile series on YouTube, Dec 15 2021.
%H Bertrand Teguia Tabuguia, <a href="https://arxiv.org/abs/2201.07127">Explicit formulas for concatenations of arithmetic progressions</a>, arXiv:2201.07127 [math.CO], 2022.
%H <a href="/index/Mo#MWP">Index entries for sequences related to Most Wanted Primes video</a>
%F a(n) = concatenate(1,2,3,...,n-2,n-1,n,n-1,n-2,...,3,2,1).
%p a:= n-> parse(cat($1..n, n-i$i=1..n-1)):
%p seq(a(n), n=1..14); # _Alois P. Heinz_, Dec 01 2021
%t Table[FromDigits[Flatten[IntegerDigits/@Join[Range[n],Reverse[Range[ n-1]]]]],{n,15}] (* _Harvey P. Dale_, Sep 02 2015 *)
%o (PARI) A173426(n)=eval(concat(vector(n*2-1,k,if(k<n,Str(k),n*2-k)))) \\ _M. F. Hasler_, Jul 29 2015
%o (Python)
%o def A173426(n): return int(''.join(str(d) for d in range(1,n+1))+''.join(str(d) for d in range(n-1,0,-1))) # _Chai Wah Wu_, Dec 01 2021
%Y This sequence and A002477 (Wonderful Demlo numbers) agree up to the 9th term.
%Y Cf. A002275, A007908, A075023, A075024, A259937, A359148.
%K nonn,base
%O 1,2
%A _Umut Uludag_, Feb 18 2010
%E More terms from and minor edits by _M. F. Hasler_, Jul 29 2015
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