A138148 -id:A138148 - cp's OEIS frontend

A269503 Largest prime factor of A138148(n).

101, 13, 137, 9091, 9901, 909091, 5882353, 52579, 333667, 9091, 99990001, 1058313049, 265371653, 909091, 2906161, 21993833369, 999999000001, 909090909090909091, 1111111111111111111, 909091, 1056689261, 549797184491917, 11111111111111111111111

Offset: 1

Altug Alkan, May 11 2016

nonn

Largest prime factor of (10^(n+1)+1)*(10^n-1)/9.

			a(4) = 9091 because largest prime factor of 111101111 is 9091.

Max Alekseyev, Table of n, a(n) for n = 1..344

Cf. A003020, A003021, A006530, A138148.

seq(max(max(numtheory:-factorset((10^n-1)/9)),
max(numtheory:-factorset(10^(n+1)+1))), n=1..30); # Robert Israel, May 11 2016

a006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
for(n=1, 25, print1(a006530((10^(2*n+1)-1)/9-10^n), ", "));

a(n) = max(A003020(n),A003021(n+1)) for n >= 2. - Robert Israel, May 11 2016

A134808 Cyclops numbers.

Original entry on oeis.org

0, 101, 102, 103, 104, 105, 106, 107, 108, 109, 201, 202, 203, 204, 205, 206, 207, 208, 209, 301, 302, 303, 304, 305, 306, 307, 308, 309, 401, 402, 403, 404, 405, 406, 407, 408, 409, 501, 502, 503, 504, 505, 506, 507, 508, 509, 601, 602, 603, 604, 605, 606

Offset: 1

Omar E. Pol, Nov 21 2007

Numbers with middle digit 0, that have only one digit 0, and the total number of digits is odd; the digit 0 represents the eye of a cyclops.

			109 is a cyclops number because 109 has only one digit 0 and this 0 is the middle digit.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video (2015).

Cf. A134809, A138131, A138148, A160717, A182809.

cyclopsQ[n_Integer, b_:10] := Module[{digitList = IntegerDigits[n, b], len, pos0s, flag}, len = Length[digitList]; pos0s = Select[Range[len], digitList[[#]] == 0 &]; flag = OddQ[len] && (Length[pos0s] == 1) && (pos0s == {(len + 1)/2}); Return[flag]]; Select[Range[0,999],cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)
Reap[Do[id=IntegerDigits[n];If[Position[id,0]=={{(Length[id]+1)/2}},Sow[n]],{n,0,10^3}]][[2,1]] (* Zak Seidov, Dec 17 2010 *)
cycQ[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]},OddQ[len] && DigitCount[ n,10,0]==1&&idn[[(len+1)/2]]==0]; Join[{0},Select[Range[ 0,700],cycQ]] (* Harvey P. Dale, Mar 07 2020 *)

a(n, {base=10}) = my (l=0); my (r=n-1); while (r >= (base-1)^(2*l), r -= (base-1)^(2*l); l++); return (base^(l+1) * ( (base^l-1)/(base-1) + if (base>2, fromdigits(digits(r \ ((base-1)^l), (base-1)), base)) ) + ( (base^l-1)/(base-1) + if (base>2, fromdigits(digits(r % ((base-1)^l), (base-1)), base)))) \\ Rémy Sigrist, Apr 29 2017

from itertools import product
def cyclops(upto=float('inf'), upton=float('inf')): # generator
  yield 0
  c, n, half_digits, pow10 = 0, 1, 0, 10
  while 100**(half_digits+1) < upto and n < upton:
    half_digits += 1
    pow10 *= 10
    for left in product("123456789", repeat=half_digits):
      left_plus_eye = int("".join(left))*pow10
      for right in product("123456789", repeat=half_digits):
        c, n = left_plus_eye + int("".join(right)), n+1
        if c <= upto and n <= upton: yield c
print([c for c in cyclops(upto=606)])
print([c for c in cyclops(upton=52)]) # Michael S. Branicky, Jan 05 2021

def is_cyclops(n, base=10):
    dg = n.digits(base) if n > 0 else [0]
    return len(dg) % 2 == 1 and dg[len(dg)//2] == 0 and dg.count(0) == 1
is_A134808 = lambda n: is_cyclops(n)
# D. S. McNeil, Dec 17 2010

A129868 Binary palindromic numbers with only one 0 bit.

Original entry on oeis.org

0, 5, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527, 140737479966719, 562949936644095

Offset: 0

Zak Seidov, May 24 2007

Binary expansion is 0, 101, 11011, 1110111, 111101111, ... (see A138148).
9 + 8a(n) = s^2 is a perfect square with s = 2^(n + 2) -1 = 3, 7, 15, 31, 63, ...
Numbers with middle bit 0, that have only one bit 0, and the total number of bits is odd.
The fractional part of the base 2 logarithm of a(n) approaches 1 as n approaches infinity.
Also called binary cyclops numbers.
Last digit of the decimal representation follows the pattern 5, 7, 9, 5, 5, 7, 9, 5, ... . - Alex Ratushnyak, Dec 08 2012

Robert Israel, Table of n, a(n) for n = 0..1630
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, video (2015)
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Base 10 analog is A134808.
Binary palindromic numbers, including repunits (or Mersenne numbers A000225) are in A006995. The sequence of binary pandigital (having both 0's and 1's) palindromic numbers begins 5, 9, 17, 21, 27, 33, 45, 51, 65, 73, ...
Cf. A006516, A138148.

[2^(2*n+1)-2^n-1: n in [0..25]]; // Vincenzo Librandi, Dec 08 2015

A129868:=n->2^(2*n + 1) - 2^n - 1: seq(A129868(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2015

(* 1st *) FromDigits[ #,2]&/@NestList[Append[Prepend[ #, 1], 1]&, {0}, 25] (* 2nd *) NestList[(1/2)(7 + 8# + Sqrt[9 + 8# ])&, 0, 22] (* both of these are from Zak Seidov *)
f[n_] := 2^(2n + 1) - 2^n - 1; Table[f@n, {n, 0, 22}] (* Robert G. Wilson v, Aug 24 2007 *)
Table[EulerE[2, 2^n], {n, 1, 60}]/2 - 1 (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
(* After running the program in A134808 *) Select[Range[0, 2^16 - 1], cyclopsQ[#, 2] &] (* Alonso del Arte, Dec 17 2010 *)
LinearRecurrence[{7, -14, 8}, {0, 5, 27}, 30] (* Vincenzo Librandi, Dec 08 2015 *)

concat(0, Vec(x*(5-8*x)/(1-7*x+14*x^2-8*x^3) + O(x^100))) \\ Altug Alkan, Dec 08 2015

def A129868(n): return ((m:=1<Chai Wah Wu, Mar 19 2024

a(n) = 2^(2n + 1) - 2^n - 1 = 2*4^n - 2^n - 1 = (2^n - 1)(2*2^n + 1).
G.f.: x*(8*x-5)/((x-1)*(2*x-1)*(4*x-1)).
Recurrences:
a(n) = (1/2)*(7 + 8*a(n - 1) + sqrt(9 + 8*a(n - 1))), a(0) = 0;
a(n) = 6*a(n - 1) - 8*a(n - 2) - 3, a(0) = 0, a(1) = 5;
a(n) = 7*a(n - 1) - 14*a(n - 2) + 8*a(n - 3), a(0) = 0, a(1) = 5, a(2) = 27.
a(n) = A006516(n+1) - 1.

A332112 a(n) = (10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

2, 121, 11211, 1112111, 111121111, 11111211111, 1111112111111, 111111121111111, 11111111211111111, 1111111112111111111, 111111111121111111111, 11111111111211111111111, 1111111111112111111111111, 111111111111121111111111111, 11111111111111211111111111111, 1111111111111112111111111111111

Offset: 0

M. F. Hasler, Feb 09 2020

a(0) = 2 is the only prime in this sequence, since all other terms factor as a(n) = R(n+1)*(10^n+1), where R(n) = (10^n-1)/9.

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332132 .. A332192 (variants with different repeated digit 3, ..., 9).
Cf. A332113 .. A332119 (variants with different middle digit 3, ..., 9).
Cf. A331860 & A331861 (indices of primes in non-palindromic variants).

Maple
```
A332112 := n -> (10^(2*n+1)-1)/9+10^n;
```

Array[ (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]

apply( {A332112(n)=10^(n*2+1)\9*1+10^n}, [0..15])

def A332112(n): return 10**(n*2+1)//9+10**n

a(n) = A138148(n) + 2*10^n = A002275(2n+1) + 10^n.
G.f.: (2 - 101*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.

Original entry on oeis.org

0, 202, 22022, 2220222, 222202222, 22222022222, 2222220222222, 222222202222222, 22222222022222222, 2222222220222222222, 222222222202222222222, 22222222222022222222222, 2222222222220222222222222, 222222222222202222222222222, 22222222222222022222222222222, 2222222222222220222222222222222

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332130 .. A332190 (variants with different repeated digit 3, ..., 9).
Cf. A332121 .. A332129 (variants with different middle digit 1, ..., 9).

A332120 := n -> 2*((10^(2*n+1)-1)/9-10^n);

Array[2 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]

apply( {A332120(n)=(10^(n*2+1)\9-10^n)*2}, [0..15])

def A332120(n): return (10**(n*2+1)//9-10**n)*2

a(n) = 2*A138148(n) = A002276(2n+1) - 2*10^n.
G.f.: 2*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: 2*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

Original entry on oeis.org

0, 909, 99099, 9990999, 999909999, 99999099999, 9999990999999, 999999909999999, 99999999099999999, 9999999990999999999, 999999999909999999999, 99999999999099999999999, 9999999999990999999999999, 999999999999909999999999999, 99999999999999099999999999999, 9999999999999990999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332180 (variants with different repeated digit 2, ..., 8).
Cf. A332191 .. A332197, A181965 (variants with different middle digit 1, ..., 8).

Maple
```
A332190 := n -> 10^(2*n+1)-1-9*10^n;
```

Array[10^(2 # + 1)-1-9*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{0,909,99099},20] (* Harvey P. Dale, May 28 2021 *)

apply( {A332190(n)=10^(n*2+1)-1-9*10^n}, [0..15])

def A332190(n): return 10**(n*2+1)-1-9*10^n

a(n) = 9*A138148(n) = A002283(2n+1) - A011557(n).
G.f.: 9*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

Original entry on oeis.org

7, 979, 99799, 9997999, 999979999, 99999799999, 9999997999999, 999999979999999, 99999999799999999, 9999999997999999999, 999999999979999999999, 99999999999799999999999, 9999999999997999999999999, 999999999999979999999999999, 99999999999999799999999999999, 9999999999999997999999999999999

Offset: 0

M. F. Hasler, Feb 08 2020

According to Kamada, n = 118 and n = 145126 are the only known indices of primes (the so-called palindromic near-repdigit or wing primes).

Patrick De Geest, Palindromic Wing Primes: (9)7(9), updated June 25, 2017.
Makoto Kamada, Factorization of 99...99799...99, updated Dec 11 2018.
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332190 .. A332196, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332117 .. A332187 (variants with different repeated digit 1, ..., 9).

Maple
```
A332197 := n -> 10^(n*2+1)-1-2*10^n;
```

Array[ 10^(2 # + 1) -1 -2*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,9],{7},PadRight[{},n,9]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{7,979,99799},20] (* Harvey P. Dale, Mar 03 2023 *)

apply( {A332197(n)=10^(n*2+1)-1-2*10^n}, [0..15])

def A332197(n): return 10**(n*2+1)-1-2*10^n

a(n) = 9*A138148(n) + 7*10^n.
G.f.: (7 + 202*x - 1100*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A138120 Concatenation of n digits 1, 2n-1 digits 0 and n digits 1.

Original entry on oeis.org

101, 1100011, 11100000111, 111100000001111, 1111100000000011111, 11111100000000000111111, 111111100000000000001111111, 1111111100000000000000011111111, 11111111100000000000000000111111111, 111111111100000000000000000001111111111

Offset: 1

Omar E. Pol, Apr 06 2008

a(n) has 4n-1 digits.

a(n) is also A147539(n) written in base 2. [Omar E. Pol, Nov 08 2008]

			n ........... a(n)
1 ........... 101
2 ......... 1100011
3 ....... 11100000111
4 ..... 111100000001111
5 ... 1111100000000011111

Index entries for linear recurrences with constant coefficients, signature (11011,-10121010,110110000,-100000000).

Cf. A138144, A138145, A138146, A138148, A138720, A138721, A138722, A138826.

Cf. A147539. [Omar E. Pol, Nov 08 2008]

a:= n-> parse(cat(1$n,0$(2*n-1),1$n)):
seq(a(n), n=1..11);  # Alois P. Heinz, Mar 03 2022

Table[FromDigits[Join[PadRight[{},n,1],PadRight[{},2n-1,0], PadRight[ {},n,1]]],{n,10}] (* or *) LinearRecurrence[{11011,-10121010,110110000,-100000000},{101,1100011,11100000111,111100000001111},10] (* Harvey P. Dale, Mar 19 2016 *)

Vec(x*(10001000*x^2-12100*x+101)/((x-1)*(10*x-1)*(1000*x-1)*(10000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

def a(n): return int("1"*n + "0"*(2*n-1) + "1"*n)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Mar 03 2022

G.f.: x*(10001000*x^2-12100*x+101) / ((x-1)*(10*x-1)*(1000*x-1)*(10000*x-1)). [Colin Barker, Sep 16 2013]

A181965 a(n) = 10^(2n+1) - 10^n - 1.

Original entry on oeis.org

8, 989, 99899, 9998999, 999989999, 99999899999, 9999998999999, 999999989999999, 99999999899999999, 9999999998999999999, 999999999989999999999, 99999999999899999999999, 9999999999998999999999999, 999999999999989999999999999, 99999999999999899999999999999, 9999999999999998999999999999999

Offset: 0

Ivan Panchenko, Apr 04 2012

n 9's followed by an 8 followed by n 9's.

See A183187 = {26, 378, 1246, 1798, 2917, ...} for the indices of primes.

Patrick De Geest, Palindromic Wing Primes: (9)8(9), updated: June 25, 2017.
Makoto Kamada, Factorization of 99...99899...99, updated Dec 11 2018.
Markus Tervooren, Factorizations of (9)w8(9)w, FactorDB.com
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. (A077794-1)/2 = A183187 (indices of primes).
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332190 .. A332197 (variants with different middle digit 0, ..., 7).

A181965 := n -> 10^(2*n+1)-1-10^n; # M. F. Hasler, Feb 08 2020

Array[10^(2 # + 1) - 1- 10^# &, 15, 0] (*  M. F. Hasler, Feb 08 2020 *)
Table[With[{c=PadRight[{},n,9]},FromDigits[Join[c,{8},c]]],{n,0,20}] (* Harvey P. Dale, Jun 07 2021 *)

apply( {A181965(n)=10^(n*2+1)-1-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

def A181965(n): return 10**(n*2+1)-1-10^n # M. F. Hasler, Feb 08 2020

From M. F. Hasler, Feb 08 2020: (Start)
a(n) = 9*A138148(n) + 8*10^n = A002283(2n+1) - A011557(10^n).
G.f.: (8 + 101*x - 1000*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2. (End)

Edited and extended to a(0) = 8 by M. F. Hasler, Feb 10 2020

A332189 a(n) = 8*(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

9, 898, 88988, 8889888, 888898888, 88888988888, 8888889888888, 888888898888888, 88888888988888888, 8888888889888888888, 888888888898888888888, 88888888888988888888888, 8888888888889888888888888, 888888888888898888888888888, 88888888888888988888888888888, 8888888888888889888888888888888

Offset: 0

M. F. Hasler, Feb 08 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different "wing" digit 1, ..., 8).
Cf. A332180 .. A332187 (variants with different middle digit 0, ..., 7).

A332189 := n -> 8*(10^(2*n+1)-1)/9+10^n;

Array[8 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]

apply( {A332189(n)=10^(n*2+1)\9*8+10^n}, [0..15])

def A332189(n): return 10**(n*2+1)//9*8+10**n

a(n) = 8*A138148(n) + 9*10^n = A002282(2n+1) + 10^n.
G.f.: (9 - 101*x - 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

cp's OEIS Frontend

A269503 Largest prime factor of A138148(n).

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A134808 Cyclops numbers.

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A129868 Binary palindromic numbers with only one 0 bit.

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A332112 a(n) = (10^(2n+1)-1)/9 + 10^n.

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A332120 a(n) = 2*(10^(2n+1)-1)/9 - 2*10^n.

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A332190 a(n) = 10^(2n+1) - 1 - 9*10^n.

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A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

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A332120 a(n) = 2(10^(2n+1)-1)/9 - 210^n.