A138131 -id:A138131 - cp's OEIS frontend

A134808 Cyclops numbers.

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

A160711 Cyclops squares: squares (A000290) that are also cyclops numbers (A134808).

Original entry on oeis.org

0, 11025, 19044, 21025, 24025, 32041, 38025, 42025, 47089, 51076, 58081, 59049, 65025, 66049, 67081, 75076, 87025, 93025, 1110916, 1140624, 1170724, 1190281, 1240996, 1270129, 1290496, 1340964, 1350244, 1380625, 1420864, 1430416

Offset: 1

Omar E. Pol, Jun 08 2009

			19044 is in the sequence because it is a square (138^2) and is also a cyclops number (odd number of digits, middle digit is the only zero).
11025 is in the sequence because it is a square (105^2) and is also a cyclops number (odd number of digits, middle digit is the only zero). - _Michael B. Porter_, Jul 09 2016

Kenny Lau, Table of n, a(n) for n = 1..20001

Cf. A000290, A134808, A134809, A134148, A138131, A153806, A160712, A160717, A239828.

Select[Range[0, 1200]^2, And[OddQ@ Length@ #, #[[Ceiling[Length[#]/2]]] == 0, Count[#, 0] == 1] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 08 2016 *)
cnQ[n_]:=Module[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1 && IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[Range[1200]^2,cnQ]] (* Harvey P. Dale, Mar 19 2018 *)

A153806 Strobogrammatic cyclops numbers.

Original entry on oeis.org

0, 101, 609, 808, 906, 11011, 16091, 18081, 19061, 61019, 66099, 68089, 69069, 81018, 86098, 88088, 89068, 91016, 96096, 98086, 99066, 1110111, 1160911, 1180811, 1190611, 1610191, 1660991, 1680891, 1690691, 1810181, 1860981

Offset: 1

Omar E. Pol, Jan 15 2009

Intersection of A000787 and A134808.

			1680891 is a member because it is the same upside down (A000787) and also a cyclops number (A134808).

Kenny Lau, Table of n, a(n) for n = 1..20000

Cf. A000787, A007597, A134808, A134809, A138131, A138148, A153807.

Select[Range[10^7], And[OddQ@ Length@#, Part[#, Ceiling[Length[#]/2]] == 0, Times @@ Boole@ Map[MemberQ[{0, 1, 6, 8, 9}, #] &, Union@ #] == 1, Count[#, 0] == 1, (Take[#, Floor[Length[#]/2]] /. {6 -> 9, 9 -> 6}) ==
Reverse@ Take[#, -Floor[Length[#]/2]]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 05 2016 *)

import sys
f = open('b153806.txt', 'w')
i = 1
n = 0
a = [""]
r = [""]  #reversed strobogrammatically
while True:
    for x,y in zip(a,r):
        f.write(str(i)+" "+x+"0"+y+"\n")
        i += 1
        if i>20000:
            f.close()
            sys.exit()
    a = sum([[x+"1",x+"6",x+"8",x+"9"] for x in a],[])
    r = sum([["1"+x,"9"+x,"8"+x,"6"+x] for x in r],[])
# Kenny Lau, Jul 05 2016

Extended beyond 11011 by R. J. Mathar, Jan 17 2009

A160717 Cyclops triangular numbers.

Original entry on oeis.org

0, 105, 406, 703, 903, 11026, 13041, 14028, 15051, 27028, 36046, 41041, 43071, 46056, 61075, 66066, 75078, 77028, 83028, 85078, 93096, 1110795, 1130256, 1160526, 1180416, 1250571, 1290421, 1330896, 1350546, 1360425, 1380291

Offset: 1

Omar E. Pol, Jun 08 2009

Triangular numbers (A000217) that are also cyclops numbers (A134808).

			105 is in the sequence since it is both a triangular number (105 = 1 + 2 + ... + 14) and a Cyclops number (number of digits is odd, and the only zero is the middle digit). - _Michael B. Porter_, Jul 08 2016

Kenny Lau, Table of n, a(n) for n = 1..20001

Cf. A000217, A134808, A134809, A138131, A138148, A153806, A160711, A160712.

count:= 1: A[1]:= 0:
for d from 1 to 3 do
  for x from 0 to 9^d-1 do
    L:= convert(x+9^d,base,9);
    X:= add((L[i]+1)*10^(i-1),i=1..d);
    for y from 0 to 9^d-1 do
      L:= convert(y+9^d,base,9);
      Y:= add((L[i]+1)*10^(i-1),i=1..d);
      Z:= Y + 10^(d+1)*X;
      if issqr(1+8*Z) then
        count:= count+1;
        A[count]:= Z;
      fi
od od od:
seq(A[i],i=1..count); # Robert Israel, Jul 08 2016

cyclopsQ[n_] := Block[{id=IntegerDigits@n,lg=Floor[Log[10,n]+1]}, Count[id,0]==1 && OddQ@lg && id[[(lg+1)/2]]==0]; lst = {0}; Do[t = n (n + 1)/2; If[ cyclopsQ@t, AppendTo[lst, t]], {n, 0, 1670}]; lst (* Robert G. Wilson v, Jun 09 2009 *)
cyclpsQ[n_]:=With[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1&&IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[ Accumulate[ Range[2000]],cyclpsQ]] (* Harvey P. Dale, Nov 05 2024 *)

More terms from Robert G. Wilson v, Jun 09 2009

Offset and b-file changed by N. J. A. Sloane, Jul 27 2016

A153807 Strobogrammatic cyclops primes.

Original entry on oeis.org

101, 16091, 1160911, 1180811, 1190611, 1690691, 1880881, 1960961, 1990661, 6110119, 6610199, 6860989, 166906991, 168101891, 169609691, 188906881, 189808681, 196906961, 199609661, 616906919, 661609199, 666101999, 668609899

Offset: 1

Omar E. Pol, Jan 15 2009

Primes in A153806.

Cf. A000787, A007597, A134808, A134809, A138131, A138148, A153806.

Extended by Ray Chandler, May 20 2009

A160712 Composite cyclops numbers (A134808).

Original entry on oeis.org

102, 104, 105, 106, 108, 201, 202, 203, 204, 205, 206, 207, 208, 209, 301, 302, 303, 304, 305, 306, 308, 309, 402, 403, 404, 405, 406, 407, 408, 501, 502, 504, 505, 506, 507, 508, 602, 603, 604, 605, 606, 608, 609, 702, 703, 704

Offset: 1

Omar E. Pol, Jun 08 2009

Cf. A134808, A134809, A134148, A138131, A153806, A160711, A160717.

Edited by Omar E. Pol, Jul 04 2009

A136098 Prime palindromic cyclops numbers.

Original entry on oeis.org

101, 16061, 31013, 35053, 38083, 73037, 74047, 91019, 94049, 1120211, 1150511, 1160611, 1180811, 1190911, 1250521, 1280821, 1360631, 1390931, 1490941, 1520251, 1550551, 1580851, 1630361, 1640461, 1660661, 1670761, 1730371

Offset: 1

Lekraj Beedassy, Mar 15 2008

Prime entries of A138131.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A082435, A138131, A134808, A134809.

f:= proc(n,d) local L,m,k,p;
  L:= convert(9^d+n,base,9);
  p:= add((1+L[d+1-i])*(10^(i-1)+10^(2*d+1-i)),i=1..d);
  if isprime(p) then p fi;
end proc:
[seq(seq(f(i,d),i=0..9^d-1),d=1..3)]; # Robert Israel, Feb 18 2018

Select[Flatten[Table[Select[Range[10^(2n), 10^(2n+1)-1], PalindromeQ[ #] && DigitCount[ #, 10, 0]==1&&IntegerDigits[#][[(IntegerLength[#]+1)/2]]==0&], {n, 3}]],PrimeQ] (* James C. McMahon, Apr 27 2025 *)

A138831 n-th perfect number minus 1, written in base 2.

Original entry on oeis.org

101, 11011, 111101111, 1111110111111, 1111111111110111111111111, 111111111111111101111111111111111, 1111111111111111110111111111111111111

Offset: 1

Omar E. Pol, Apr 08 2008, Apr 09 2008, Apr 14 2008

Subset of A138148, cyclops numbers with binary digits, only.
Subset of A002113, palindromes in base 10.
a(n) has 2*A090748(n) digits 1.
The number of digits of a(n) is 2*A000043(n)-1, equal to A133033(n), the number of proper divisors of n-th perfect number.
a(n) = (A135627(n) written in base 2).

			n ... A000396(n) - 1 = A135627(n) ............. a(n)
1 ............ 6 - 1 = ...... 5 ............... 101
2 ........... 28 - 1 = ..... 27 .............. 11011
3 .......... 496 - 1 = .... 495 ............ 111101111
4 ......... 8128 - 1 = ... 8127 .......... 1111110111111
5 ..... 33550336 - 1 = 33550335 .... 1111111111110111111111111

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A090748, A133033, A134808, A135627, A138131, A138148.

a(n) = A138148(A090748(n)).

A160725 Cyclops semiprimes.

Original entry on oeis.org

106, 201, 202, 203, 205, 206, 209, 301, 302, 303, 305, 309, 403, 407, 501, 502, 505, 703, 706, 707, 802, 803, 807, 901, 905, 11013, 11014, 11015, 11017, 11019, 11021, 11023, 11029, 11031, 11035, 11038, 11041, 11042, 11051, 11053

Offset: 1

Omar E. Pol, Jun 12 2009

Cyclops numbers (A134808) that are also semiprimes (A001358).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A134808, A134809, A138131, A138148, A153806, A153807, A160711, A160712, A160717.

g:= proc(x,n)
  local L,i;
  L:= convert(x+9^(2*n),base,9);
  add((L[i]+1)*10^(i-1),i=1..n)+add((L[i]+1)*10^i,i=n+1..2*n)
end proc:
select(t -> numtheory:-bigomega(t)=2,[seq(seq(g(i,n),i=0..9^(2*n)-1),n=1..2)]); # Robert Israel, Jan 20 2019

Select[Range@ 12000, And[OddQ@ #2, #3[[Ceiling[#2/2] ]] == 0, Count[#3, 0] == 1, PrimeOmega@ #1 == 2] & @@ {#, IntegerLength@ #, IntegerDigits@ #} &] (* or *)
Select[Flatten@ Table[a (10^(d + 1)) + b, {d, 2}, {a, FromDigits /@ Tuples[Range@ 9, {d}]}, {b, FromDigits /@ Tuples[Range@ 9, {d}]}], PrimeOmega@ # == 2 &] (* Michael De Vlieger, Jan 20 2019 *)

cp's OEIS Frontend

A134808 Cyclops numbers.

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A160711 Cyclops squares: squares (A000290) that are also cyclops numbers (A134808).

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A153806 Strobogrammatic cyclops numbers.

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A160717 Cyclops triangular numbers.

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A153807 Strobogrammatic cyclops primes.

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A160712 Composite cyclops numbers (A134808).

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A136098 Prime palindromic cyclops numbers.

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A138831 n-th perfect number minus 1, written in base 2.

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A160725 Cyclops semiprimes.