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A173426
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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.
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38
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1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345678910987654321, 123456789101110987654321, 1234567891011121110987654321, 12345678910111213121110987654321, 123456789101112131413121110987654321
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OFFSET
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1,2
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COMMENTS
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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
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.
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*n-28) - 2*10^(2*n-9) + 8790000000121)/99^2, for 10 <= n < 10^2;
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
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REFERENCES
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D. Broadhurst, Primes from concatenation: results and heuristics, Number Theory List, Aug 01 2015 and later postings.
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LINKS
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FORMULA
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a(n) = concatenate(1,2,3,...,n-2,n-1,n,n-1,n-2,...,3,2,1).
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MAPLE
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a:= n-> parse(cat($1..n, n-i$i=1..n-1)):
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MATHEMATICA
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Table[FromDigits[Flatten[IntegerDigits/@Join[Range[n], Reverse[Range[ n-1]]]]], {n, 15}] (* Harvey P. Dale, Sep 02 2015 *)
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PROG
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(PARI) A173426(n)=eval(concat(vector(n*2-1, k, if(k<n, Str(k), n*2-k)))) \\ M. F. Hasler, Jul 29 2015
(Python)
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
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CROSSREFS
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This sequence and A002477 (Wonderful Demlo numbers) agree up to the 9th term.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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More terms from and minor edits by M. F. Hasler, Jul 29 2015
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STATUS
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approved
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