A153806 -id:A153806 - cp's OEIS frontend

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

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 *)

A332180 a(n) = 8(10^(2n+1)-1)/9 - 810^n.

Original entry on oeis.org

0, 808, 88088, 8880888, 888808888, 88888088888, 8888880888888, 888888808888888, 88888888088888888, 8888888880888888888, 888888888808888888888, 88888888888088888888888, 8888888888880888888888888, 888888888888808888888888888, 88888888888888088888888888888, 8888888888888880888888888888888

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 only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332181 .. A332189 (variants with different middle digit 1, ..., 9).
Subsequence of A006072 (numbers with mirror symmetry about middle), A153806 (strobogrammatic cyclops numbers), and A204095 (numbers whose decimal digits are in {0,8}).

A332180 := n -> 8*((10^(2*n+1)-1)/9-10^n);

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

apply( {A332180(n)=(10^(n*2+1)\9-10^n)*8}, [0..15])

def A332180(n): return (10**(n*2+1)//9-10**n)*8

a(n) = 8*A138148(n) = A002282(2n+1) - 8*10^n.
G.f.: 8*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.: 8*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

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

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 *)

A246880 6((10^n-1)/9)(10^(n+1))+9*(10^n-1)/9.

Original entry on oeis.org

0, 609, 66099, 6660999, 666609999, 66666099999, 6666660999999, 666666609999999, 66666666099999999, 6666666660999999999, 666666666609999999999, 66666666666099999999999, 6666666666660999999999999, 666666666666609999999999999, 66666666666666099999999999999

Offset: 0

Felix Fröhlich, Sep 06 2014

Numbers of the form 6...609...9 (i.e., consisting of an odd number of digits with the middle digit 0, all digits to the left of the middle digit 6 and all digits to the right of the middle digit 9).

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

Subsequence of A001743, A039434, A111065, A111156, A153806, A169731 and A227793.

[(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9) : n in [0..15]]; // Wesley Ivan Hurt, Sep 15 2014

A246880:=n->(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9): seq(A246880(n),n=0..15); # Wesley Ivan Hurt, Sep 15 2014

Table[(6*((10^n - 1)/9))*(10^(n + 1)) + (9*(10^n - 1)/9), {n, 15}] (* Wesley Ivan Hurt, Sep 15 2014 *)
Join[{0}, CoefficientList[Series[20/(3 x - 300 x^2) + 1/(x^2 - x) + 17/(30 x^2 - 3 x), {x, 0, 30}], x]] (* Wesley Ivan Hurt, Sep 15 2014 *)

a(n)=6*((10^n-1)/9)*(10^(n+1))+9*(10^n-1)/9

G.f.: 20/(3-300*x)+1/(x-1)+17/(30*x-3); a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3). - Wesley Ivan Hurt, Sep 15 2014

A287092 Strobogrammatic nonpalindromic numbers.

Original entry on oeis.org

69, 96, 609, 619, 689, 906, 916, 986, 1691, 1961, 6009, 6119, 6699, 6889, 6969, 8698, 8968, 9006, 9116, 9696, 9886, 9966, 16091, 16191, 16891, 19061, 19161, 19861, 60009, 60109, 60809, 61019, 61119, 61819, 66099, 66199, 66899, 68089, 68189, 68889, 69069, 69169, 69869, 86098, 86198, 86898, 89068, 89168

Offset: 1

Ilya Gutkovskiy, May 19 2017

Nonpalindromic numbers which are invariant under a 180-degree rotation.
Numbers that are the same upside down and containing digits 6, 9.
Intersection of A000787 and A029742.
Union of this sequence and A006072 gives A000787.

Cf. A000787, A007597, A018848, A029742, A006072, A153806, A169731.

fQ[n_] := Block[{s = {0, 1, 6, 8, 9}, id = IntegerDigits[n]}, If[ Union[ Join[s, id]] == s && (id /. {6 -> 9, 9 -> 6}) == Reverse[id], True, False]]; Select[ Range[0, 89168], fQ[ # ] && ! PalindromeQ[ # ] &]

cp's OEIS Frontend

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

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A332180 a(n) = 8*(10^(2n+1)-1)/9 - 8*10^n.

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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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A160725 Cyclops semiprimes.

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A246880 6*((10^n-1)/9)*(10^(n+1))+9*(10^n-1)/9.

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Magma

Maple

Mathematica

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A287092 Strobogrammatic nonpalindromic numbers.

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Mathematica

A332180 a(n) = 8(10^(2n+1)-1)/9 - 810^n.

A246880 6((10^n-1)/9)(10^(n+1))+9*(10^n-1)/9.