A173426 -id:A173426 - cp's OEIS frontend

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.

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

A075024 a(n) is the largest prime divisor of the number A173426(n) = concatenate(1,2,...,n-1,n,n-1,...,2,1); a(1) = 1.

Original entry on oeis.org

1, 11, 37, 101, 271, 37, 4649, 137, 333667, 12345678910987654321, 17636684157301569664903, 2799473675762179389994681, 2354041513534224607850261, 2068140300159522133, 498056174529497, 112240064764214229701, 4188353169004802474320231191377

Offset: 1

Amarnath Murthy, Sep 01 2002

Also for 1 < n < 10, a(n) is the common prime divisor for all A010785(m) which consist of n digits. - Alexander R. Povolotsky, Jun 05 2014, corrected by M. F. Hasler, Jul 30 2015

According to the definition (and given terms), this is the greatest prime factor (A006530) of A173426 and not of A002477, as an earlier formula asserted and which may have been an assumption of the preceding comment. - M. F. Hasler, Jul 29 2015

			a(5) = 271 as 123454321 = 41*41*271*271.
a(25) = 12471243489559387823527232424981012432152516319410549 is the larger factor of the semiprime A173426(24) = A075023(25) * a(n).

Amiram Eldar, Table of n, a(n) for n = 1..58 (terms 1..36 from M. F. Hasler)
FactorDB, (121*10^(4*n-19) - 1002*10^(4*n-28) - 2*10^(2*n-9) + 879*10^10 + 121)/99^2.

Cf. A002477, A006530, A010785, A075019, A075020, A075021, A075022, A075023, A173426.

Table[FactorInteger[FromDigits[Join[Flatten[IntegerDigits/@Range[ n]], Flatten[ IntegerDigits/@Range[n-1,1,-1]]]]][[-1,1]],{n,20}] (* Harvey P. Dale, May 20 2016 *)

a(n) = {if (n == 1, return (1)); s = ""; for (i=1, n, s = concat(s, Str(i));); forstep (i=n-1, 1, -1, s = concat(s, Str(i));); f = factor(eval(s)); f[#f~, 1];} \\ Michel Marcus, Jun 05 2014

A075024(n)=A006530(A173426(n)) \\ A006530 should provide efficient code and also covers the case n=1. - M. F. Hasler, Jul 29 2015

a(n) = A006530(A173426(n)). - Michel Marcus, Jun 05 2014, corrected by M. F. Hasler, Jul 29 2015

More terms from Sascha Kurz, Jan 03 2003
a(16)-a(17) from Michel Marcus, Jun 05 2014
More terms 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]

A260597 Primes as they occur for the first time as factors of terms of A173426 = concatenation(1,2,...,n,n-1,...,1).

Original entry on oeis.org

11, 3, 37, 101, 41, 271, 7, 13, 239, 4649, 73, 137, 333667, 12345678910987654321, 17636684157301569664903, 2799473675762179389994681, 1109, 4729, 2354041513534224607850261, 571, 3167, 10723, 439781, 2068140300159522133, 75401, 687437, 759077450603

Offset: 1

M. F. Hasler, Jul 29 2015

nonn

Or, distinct elements of A260589 in the order they occur for the first time.

			A173426(1) = 1;  A173426(2) = 121 = 11^2 => a(1) = 11.
A173426(3) = 12321 = 3^2 37^2 => a(2..3) = (3, 37).
A173426(4) = 1234321 = 11^2 101^2 => a(4) = 101.
A173426(5) = 123454321 = 41^2 271^2 => a(5..6) = (41, 271).
A173426(6) = 12345654321 = 3^2 7^2 11^2 13^2 37^2 => a(7..8) = (7, 13).

M. F. Hasler and Chai Wah Wu, Table of n, a(n) for n = 1..114 (a(n) for n = 1..84 from M. F. Hasler)

S=[];apply(t->S=setunion(S,t=setminus(Set(t),S));t, vector(30,n,A260589_row(n)))

from sympy import primefactors
A260597_list = []
for n in range(1,10):
    m = primefactors(int(''.join([str(d) for d in range(1,n+1)]+[str(d) for d in range(n-1,0,-1)])))
    for p in m:
        if not p in A260597_list:
            A260597_list.append(p) # Chai Wah Wu, Aug 11 2015

A359148 1, together with numbers k such that A173426(k) is prime.

Original entry on oeis.org

1, 10, 2446

Offset: 1

N. J. A. Sloane, Feb 17 2023

Many of the comments following the "Most Wanted Prime" video assumed that A173426(1) was 11, and so "Numbers k such that A173426(k) is prime" should begin 1, 10, 2446, ... This is incorrect, A173426(1) = 1 and is not prime.
However, if the 1 is omitted the sequence is too short to include in the OEIS, so the present sequence will serve as a place-holder until we find the next term.
Its inclusion could also be justified by the OEIS policy of including published but erroneous sequences to serve as pointers to the correct versions.
Serge Batalov comments (see A173426) that a(4) >= 60000.

Brady Haran and N. J. A. Sloane, Most Wanted Prime, Numberphile video, December 2021.
Index entries for sequences related to Most Wanted Primes video

Cf. A173426, A360503.

A362680 a(n) is the number of decimal digits in A173426(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 1

David Cleaver, Apr 29 2023

			a(12)=28 since 1234567891011121110987654321 has 28 digits.

Stefano Spezia, Table of n, a(n) for n = 1..5000

Cf. A173426, A058183, A055642.

a[n_]:=IntegerLength[FromDigits[Flatten[IntegerDigits/@Join[Range[n], Reverse[Range[n-1]]]]]]; Array[a,63] (* Stefano Spezia, Apr 16 2025 *)

a(n)={my(t=logint(n,10)+1); 2*n*t-2*(10^t-1)/9+t}

def a(n): return ((n*(t:=len(str(n)))-(10**t-1)//9)<<1) + t
print([a(n) for n in range(1, 64)]) # Michael S. Branicky, May 02 2023

a(n) = A058183(n) + A058183(n-1), for n >= 2.
a(n) = A055642(A173426(n)).
a(n) = 2*A058183(n) - A055642(n).

A260853 Base-3 representation of a(n) is the concatenation of the base-3 representations of 1, 2, ..., n, n-1, ..., 1.

Original entry on oeis.org

0, 1, 16, 439, 35350, 2864599, 232046890, 18795930559, 1522471570630, 369960528437035, 269701223137448146, 196612191672080116867, 143330287729139571972130, 104487779754548024866115515, 76171591441065652665051372946, 55529090160536864641400481743827

Offset: 0

M. F. Hasler, Aug 01 2015

The base 3 is listed in A260343, which means that a(3) = A260851(3) = 439 = 121021_3 is prime and therefore in A260852. See these sequences for more information.

			a(0) = 0 is the result of the empty sum corresponding to 0 digits.
a(2) = 16 = 121_3 is the concatenation of (1, 2, 1).
a(3) = 439 = 121021_3 is the concatenation of (1, 2, 10, 2, 1), where the middle "10" is the base-3 representation of 3.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015
Index entries for sequences related to Most Wanted Primes video

Base-3 variant of A173426 (base 10) and A173427 (base 2). See A260854 - A260866 for variants in other bases b = 4, ..., 16.

Join[{0},Table[FromDigits[Join[Flatten[IntegerDigits[Range[n], 3]], Flatten[ Reverse[ Most[ IntegerDigits[Range[n],3]]]]],3],{n,20}]] (* Harvey P. Dale, Mar 11 2019 *)

a(n,b=3)=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i))

A260343 Numbers n such that the base-n number formed by concatenating the base-n numbers 1 2 ... n-1 n n-1 ... 2 1 is prime.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 16, 40, 104, 8840

Offset: 1

N. J. A. Sloane, Aug 01 2015

n = 8840 only corresponds to a probable prime (with 69770 decimal digits).

The concatenation (in base n) of the base-n numbers 1 2 3 ... k-1 k k-1 ... 3 2 1 is a square for k

Sequence A260852 lists the actual primes, A260852(k) = A260851(a(k)). - M. F. Hasler, Aug 02 2015

			For n = 2 we get the binary number 1 10 1 = 1101 = 13 (in decimal).
For n = 10 we get (as David Broadhurst remarks) the "memorable" decimal prime 12345678910987654321.
For n = 16 the prime is the hexadecimal number 123456789abcdef10fedcba987654321.

D. Broadhurst, Primes from concatenation: results and heuristics, NmbrThry List, August 1, 2015

For n=2 see A173427, for n=10 see A173426.

For n=3 through n=16 see A260853 - A260866.

Select[Range[2, 120], PrimeQ@ Total[Times @@@ Transpose[{(Function[p, Power[#, p]] /@ Reverse@ Delete[Range[0, 2 # - 1], # + 1]), Flatten@ {Range[#], Reverse@ Range[# - 1]}}]] &] (* Michael De Vlieger, Aug 02 2015 *)
bnpQ[n_]:=PrimeQ[FromDigits[Flatten[Join[IntegerDigits[#,n]&/@Range[n], IntegerDigits[ #,n]&/@Reverse[Range[n-1]]]],n]]; Select[Range[2,110],bnpQ] (* Harvey P. Dale, Feb 26 2023 *)

for(b=2,9e9,ispseudoprime(p=(1+b*c=(b^b-1)\(b-1))*(c-b+1)-1)&&print1(b", ")); \\ D. Broadhurst and M. F. Hasler, Aug 02 2015

from gmpy2 import is_prime
def intbase(dlist,b=10): # convert list of digits in base b to integer
    y = 0
    for d in dlist:
        y = y*b + d
    return y
A260343_list = [n for n in range(2,500) if is_prime(intbase(list(range(1,n))+[1,0]+list(range(n-1,0,-1)), n))] # Chai Wah Wu, Aug 01 2015

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cp's OEIS Frontend

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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A075024 a(n) is the largest prime divisor of the number A173426(n) = concatenate(1,2,...,n-1,n,n-1,...,2,1); 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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A260597 Primes as they occur for the first time as factors of terms of A173426 = concatenation(1,2,...,n,n-1,...,1).

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A359148 1, together with numbers k such that A173426(k) is prime.

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A362680 a(n) is the number of decimal digits in A173426(n).

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