A075024 -id:A075024 - cp's OEIS frontend

A173426 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.

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

Offset: 1

Umut Uludag, Feb 18 2010

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.
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*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
Curiously, 1234567891010987654321 is also a prime (see A259937). - N. J. A. Sloane, Nov 30 2021

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

G. C. Greubel, Table of n, a(n) for n = 1..150
FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.
Shyam Sunder Gupta, Puzzle 794. Prime Generalized Palindromes, The Prime Puzzles and Problems Connection.
S. S. Gupta, A new 17350 digit Symmetric Prime, NmbrThry List, July 20, 2015.
Brady Haran and N. J. A. Sloane, The Most Wanted Prime Number, Numberphile series on YouTube, Dec 15 2021.
Bertrand Teguia Tabuguia, Explicit formulas for concatenations of arithmetic progressions, arXiv:2201.07127 [math.CO], 2022.
Index entries for sequences related to Most Wanted Primes video

This sequence and A002477 (Wonderful Demlo numbers) agree up to the 9th term.

Cf. A002275, A007908, A075023, A075024, A259937, A359148.

a:= n-> parse(cat($1..n, n-i$i=1..n-1)):
seq(a(n), n=1..14);  # Alois P. Heinz, Dec 01 2021

Table[FromDigits[Flatten[IntegerDigits/@Join[Range[n],Reverse[Range[ n-1]]]]],{n,15}] (* Harvey P. Dale, Sep 02 2015 *)

A173426(n)=eval(concat(vector(n*2-1,k,if(kM. F. Hasler, Jul 29 2015

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

a(n) = concatenate(1,2,3,...,n-2,n-1,n,n-1,n-2,...,3,2,1).

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

A005422 Largest prime factor of 10^n - 1.

Original entry on oeis.org

3, 11, 37, 101, 271, 37, 4649, 137, 333667, 9091, 513239, 9901, 265371653, 909091, 2906161, 5882353, 5363222357, 333667, 1111111111111111111, 27961, 10838689, 513239, 11111111111111111111111, 99990001, 182521213001, 1058313049

Offset: 1

N. J. A. Sloane

nonn

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Table of n, a(n) for n = 1..352
K. Beschorner, Factorizations of Repunit Numbers of the form (10^n-1)/9 (111111...). [Retrieved July 30, 2015]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
Yousuke Koide, Factorizations of Repunit Numbers.
S. S. Wagstaff, Jr., The Cunningham Project

Same as A003020 except for the additional a(1) = 3.
Cf. A002275, A002283, A004023, A006530, A067063, A075024, A102380.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(10^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

A005422 := proc(n)
    10^n-1 ;
    A006530(%) ;
end proc: # R. J. Mathar, Dec 02 2016

Table[FactorInteger[10^n - 1][[-1, 1]], {n, 1, 40}] (* Vincenzo Librandi, Jul 13 2016 *)

a(n)=vecmax(factor(10^n-1)[,1]) \\ Simplified by M. F. Hasler, Jul 30 2015

For n > 1, a(n) = A003020(n). For 1 < n < 10, a(n) = A075024(n). - M. F. Hasler, Jul 30 2015
a(n) = A006530(A002283(n)). - Vincenzo Librandi, Jul 13 2016
a(A004023(n)) = A002275(A004023(n)). - Bernard Schott, May 24 2022

Terms to a(100) in b-file from Yousuke Koide added by T. D. Noe, Dec 06 2006
Edited by M. F. Hasler, Jul 30 2015
a(101)-a(322) in b-file from Ray Chandler, Apr 22 2017
a(323)-a(352) in b-file from Max Alekseyev, Apr 26 2022

A075023 a(n) = the smallest prime divisor of A173426(n) = concatenation of (1, 2, 3,..., n, n-1, ..., 1) for n > 1; a(1) = 1.

Original entry on oeis.org

1, 11, 3, 11, 41, 3, 239, 11, 3, 12345678910987654321, 7, 3, 1109, 7, 3, 71, 7, 3, 251, 7, 3, 70607, 7, 3, 989931671244066864878631629, 7, 3, 149, 7, 3, 827, 7, 3, 197, 7, 3, 39907897297, 7, 3, 17047, 7, 3, 191, 7, 3, 967, 7, 3, 139121, 7, 3, 109, 7, 3, 5333, 7, 3

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(5) = 41 as 123454321 = 41*41*271*271.
a(25) = 989931671244066864878631629 is the smaller factor of the semiprime A173426(24) = a(25) * A075023(25).
A173426(37) = 39907897297 * P58 * P59, where Pxx are primes with xx digits, therefore a(37) = 39907897297.

FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A075019, A075020, A075021, A075022, A075024.

A075023(n)=A020639(A173426(n)) \\ Efficient code for computing the least prime factor should be developed in A020639. For n = 37, use \g3 (debugging level 3) to see the lpf within milliseconds, while factorization would take hours. - M. F. Hasler, Jul 29 2015

a(n) = A020639(A173426(n)). a(3n) = 3 for all n > 0. a(3n-1) = 7 for 3 < n < 34. - M. F. Hasler, Jul 29 2015

More terms from Sascha Kurz, Jan 03 2003

Terms beyond a(24) from M. F. Hasler, Jul 29 2015

A260587 Number of distinct prime factors of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

Original entry on oeis.org

0, 1, 2, 2, 2, 5, 2, 4, 3, 1, 2, 3, 3, 6, 6, 4, 4, 4, 6, 2, 5, 3, 4, 8, 2, 6, 8, 2, 4, 9, 4, 9, 6, 6, 6, 7, 3, 5, 7, 4, 6, 6, 3, 6

Offset: 1

M. F. Hasler, Jul 29 2015

			a(2) = 1 since A173426(2) = 121 = 11*11 has only one distinct prime factor, 11.
a(21) = 5 since A173426(21) = 3^2 * 7 * 828703 * 94364768151913037621 * 250591098443370396365457961250972909.
a(25) = 2 since A173426(25) = A075023(n) * A075024(n) is a semiprime.

FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A001221.
See A260588 for the variant where prime factors are counted with multiplicity.
See also A075023 and A075024 for the smallest and largest prime factor of the terms.

PARI
```
a(n)=omega(A173426(n))
```

a(38)-a(44) added using factordb.com by Jinyuan Wang, Mar 05 2020

A260588 Number of prime factors, counted with multiplicity, of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

Original entry on oeis.org

0, 2, 4, 4, 4, 10, 4, 8, 8, 1, 2, 5, 3, 6, 7, 4, 5, 8, 6, 2, 6, 3, 4, 9, 2, 6, 11, 2, 4, 10, 4, 9, 8, 6, 6, 12, 3, 6, 8, 4, 6, 7, 3, 6

Offset: 1

M. F. Hasler, Jul 29 2015

			a(2) = 2 since A173426(2) = 121 = 11*11 has twice the factor 11.
a(21) = 6 since A173426(21) = 3 * 3 * 7 * 828703 * 94364768151913037621 * 250591098443370396365457961250972909.

FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

See A260587 for the variant where only distinct prime factors are counted.
See also A075023 and A075024 for the smallest and largest prime factor of the terms.
Cf. A001222.

PARI
```
a(n)=bigomega(A173426(n))
```

a(n) = A001222(A173426(n)).

Terms beyond a(30) via factorization results by Serge Batalov, added by M. F. Hasler, Jul 30 2015

a(38)-a(44) added using factordb.com by Jinyuan Wang, Mar 05 2020

A260589 Irregular table read by rows: n-th row lists the prime factors of A173426(n), with repetition.

Original entry on oeis.org

11, 11, 3, 3, 37, 37, 11, 11, 101, 101, 41, 41, 271, 271, 3, 3, 7, 7, 11, 11, 13, 13, 37, 37, 239, 239, 4649, 4649, 11, 11, 73, 73, 101, 101, 137, 137, 3, 3, 3, 3, 37, 37, 333667, 333667, 12345678910987654321, 7, 17636684157301569664903, 3, 3, 7, 7, 2799473675762179389994681, 1109, 4729

Offset: 1

M. F. Hasler, Jul 29 2015

Row lengths are given by A260588(n). In particular, row n = 1 would have length 0, i.e., no element, because A173426(1) = 1 has no prime factors. Therefore the sequence can be considered to start with row n = 2. (The offset refers to the k-th element of the "flattened" sequence.)

For n = 1 through n = 9, A173426(n) is the square of the repunit 1...1 of length n, therefore every prime factor appears twice. This is no longer the case for n > 9.

M. F. Hasler, Table of k, a(k) for k = 1..150 (Rows n = 2 through 30, flattened.)
M. F. Hasler, Factorization of A173426 = 123...321, OEIS wiki, July 2015.

Cf. A001222, A075023, A075024, A173426, A260587, A260588.

PARI
```
A260589_row(n)=A027746_row(A173426(n))
```

vector(30,n,A027746_row(A173426(n))) \\ You may concat() this.

  n |  A173426(n) | factors = n-th row of this table
|      1      | []
|     121     | [11, 11]
|    12321    | [3, 3, 37, 37]
|   1234321   | [11, 11, 101, 101]
|  123454321  | [41, 41, 271, 271]
| 12345654321 | [3, 3, 7, 7, 11, 11, 13, 13, 37, 37]

cp's OEIS Frontend

A173426 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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A005422 Largest prime factor of 10^n - 1.

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A075023 a(n) = the smallest prime divisor of A173426(n) = concatenation of (1, 2, 3,..., n, n-1, ..., 1) for n > 1; a(1) = 1.

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A260587 Number of distinct prime factors of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

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A260588 Number of prime factors, counted with multiplicity, of A173426(n) = concatenation of (1, 2, ..., n, n-1, ..., 1).

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A260589 Irregular table read by rows: n-th row lists the prime factors of A173426(n), with repetition.

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Keywords

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Programs

PARI

PARI

Formula