A134808 -id:A134808 - cp's OEIS frontend

A136098 Prime palindromic cyclops numbers.

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 *)

A182809 Cyclops-Fibonacci numbers.

Original entry on oeis.org

0, 75025, 6557470319842, 14472334024676221, 99194853094755497

Offset: 1

Omar E. Pol, Dec 16 2010

Note that a(5) = 99194853094755497 is the only known Cyclops-Fibonacci prime.
Next term, if it exists, is > Fibonacci(2359000). - Lars Blomberg, May 10 2011
This sequence is similar to A182811 in the sense that both have four positive terms and the only known prime is also the largest known term. - Omar E. Pol, Feb 18 2018
Indices in A000045 are 0, 25, 63, 79, 83. - Michel Marcus and Omar E. Pol, Feb 18 2018

			a(2) = 75025 is in the sequence because 75025 is a Fibonacci number A000045 and 75025 is also a cyclops number A134808.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 99194853094755497

Intersection of A000045 and A134808.

Cf. A134809, A138148, A160717, A182811.

(* First run the program given for A134808 *) Select[Fibonacci[Range[10^3]], cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)

a(4) inserted by Alois P. Heinz, Dec 16 2010

A239827 Cyclops numbers whose squares are cyclops numbers.

Original entry on oeis.org

0, 105, 205, 305, 11014, 11023, 11041, 11059, 12017, 12021, 12046, 12075, 12079, 13027, 13031, 13096, 14011, 14018, 14043, 14068, 14075, 14082, 16019, 16022, 16044, 16072, 16075, 17012, 17091, 17094, 18014, 18039, 18075, 18086, 19016, 19029, 19037, 19058

Offset: 1

Colin Barker, Mar 27 2014

Subsequence of A239589.

			12046 is in the sequence because 12046^2 = 145106116, and both 12406 and 145106116 are cyclops numbers.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A134808, A239589, A239828.

is_cyclops(k) = {
  if(k==0, return(1));
  my(d=digits(k), j);
  if(#d%2==0 || d[#d\2+1]!=0, return(0));
  for(j=1, #d\2, if(d[j]==0, return(0)));
  for(j=#d\2+2, #d, if(d[j]==0, return(0)));
  return(1)}
s=[]; for(n=0, 100000, if(is_cyclops(n) && is_cyclops(n^2), s=concat(s, n))); s

a(n) = sqrt(A239828(n)).

A252480 Numbers whose decimal representation has at least one '0' digit in a position other than the final digit.

Original entry on oeis.org

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 600, 601, 602, 603, 604, 605, 606

Offset: 1

M. F. Hasler, Dec 28 2014

Similar but different sequences are the "Cyclops numbers" A134808 and A032945 and A051022, which are subsequences, except for the 1- and 2-digit terms.
Also, numbers whose decimal representation cannot be split up between any two digits without producing a string with a leading zero (other than "0" itself).
Also, numbers n > 9 such that floor(n/10) is in A011540, i.e., has a digit '0'.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Select[Range[10,700],DigitCount[Floor[#/10],10,0]>0&] (* Harvey P. Dale, May 10 2020 *)

PARI
```
is(n)=n>9 && !vecmin(digits(n\10))
```

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)).

A160561 Cyclops primes with circular digits {0,6,8,9}.

Original entry on oeis.org

809, 66089, 68099, 86069, 88069, 89069, 99089, 6680689, 6680699, 6680969, 6690689, 6690899, 6690989, 6860869, 6860989, 6860999, 6890699, 6890969, 6960869, 6980669, 6980899, 6980969, 6990889, 8660689, 8660699, 8660969

Offset: 1

Ki Punches, May 19 2009

The sequence is probably infinite.
The sequence A134809 restricted to cases with digits 6, 8 or 9 (see A001743) at the off-center positions.
Primes in A274765. - Omar E. Pol, Jul 06 2016
Each term is equal to 9 mod 10. - Harvey P. Dale, Feb 02 2021

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

Intersection of A000040 (primes), A001743 (numbers with circular digits) and A134808 (cyclops numbers).
Also intersection of A043580 (primes with circular digits) and A134809 (cyclops primes).
Cf. A134808, A134809, A153807, A156200, A274765.

Select[Prime@ Range[10^6], And[OddQ@ Length@ #, Times @@ Boole@ Map[MemberQ[{0, 6, 8, 9}, #] &, Union@ #] == 1, Part[#, Ceiling[Length[#]/2]] == 0, Count[#, 0] == 1] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 05 2016 *)
Table[Select[FromDigits/@(Flatten[Join[{Take[#,Length[#]/2],0,Take[#,-Length[#]/2]}]]&/@Tuples[{6,8,9},n]),PrimeQ],{n,2,6,2}]//Flatten (* Harvey P. Dale, Feb 02 2021 *)

Edited and corrected by Ray Chandler and R. J. Mathar, May 20 2009

Definition simplified by Omar E. Pol, Jun 05 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 *)

A162198 Even cyclops numbers.

Original entry on oeis.org

0, 102, 104, 106, 108, 202, 204, 206, 208, 302, 304, 306, 308, 402, 404, 406, 408, 502, 504, 506, 508, 602, 604, 606, 608, 702, 704, 706, 708, 802, 804, 806, 808, 902, 904, 906, 908, 11012, 11014, 11016, 11018, 11022, 11024, 11026, 11028, 11032, 11034, 11036

Offset: 1

Omar E. Pol, Jul 04 2009

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A134808, A134809.

Join[{0},Select[Range[14000],OddQ[IntegerLength[#]]&&EvenQ[#] && DigitCount[ #,10,0] == 1&&IntegerDigits[#][[(IntegerLength[ #]+1)/2]] == 0&]] (* Harvey P. Dale, May 03 2017 *)

Corrected by Harvey P. Dale, May 03 2017

A182811 Cyclops-Lucas numbers.

Original entry on oeis.org

64079, 1860498, 4870847, 688846502588399

Offset: 1

Omar E. Pol, Dec 20 2010

a(4) = 688846502588399 is the only known Cyclops-Lucas prime.
It seems likely that these four are the only terms. There are no further terms below Lucas(10^7), and that number in decimal contains 208435 zeros (with ~208988 expected assuming normality), whereas a member of this sequence must have only 1. - D. S. McNeil, Dec 21 2010
This sequence is similar to A182809 in the sense that both have four positive terms and the only known prime is also the largest known term. - Omar E. Pol, Dec 21 2010
Indices in A000032 are 23, 30, 32, 71. - Michel Marcus and Omar E. Pol, Feb 18 2018

			a(1) = 64079 is in the sequence because 64079 is a Lucas number and it is also a cyclops number.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 688846502588399

Intersection of A000032 and A134808.

Cf. A005479, A134809, A182809.

(* First run the program given for A134808 *) Select[LucasL[Range[10^3]], cyclopsQ] (* Alonso del Arte, Dec 20 2010 *)
Select[LucasL[Range[500]],OddQ[IntegerLength[#]]&&DigitCount[#,10,0]==1&&IntegerDigits[#][[(IntegerLength[#]+1)/2]]==0&] (* Harvey P. Dale, Jul 01 2017 *)

Intersection of A000032 and A134808.

A183058 Cyclops Sophie-Germain primes.

Original entry on oeis.org

509, 809, 12011, 12041, 13049, 14081, 16091, 18041, 21011, 21089, 22013, 22079, 23099, 25073, 28019, 29021, 29033, 31019, 33023, 33053, 35069, 35081, 35099, 36083, 37013, 37049, 38039, 39089, 41081, 42023, 42071, 42089, 43013

Offset: 1

Omar E. Pol, Dec 26 2010

Sophie Germain primes which are also Cyclops numbers.

			509 is in the sequence because 509 is a Sophie Germain prime A005384 and it is also a Cyclops number A134808.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A005384, A134808, A134809, A136098, A153807, A183059.

isA005384 := proc(n) isprime(n) and isprime(2*n+1) ; end proc:
isA134808 := proc(n) local dgs,ndgs; dgs := convert(n,base,10) ; mdg := (nops(dgs)+1)/2 ; if type(nops(dgs),'even') then false; elif n = 0 then true; else if op(mdg,dgs) <> 0 then false; else if mul(op(k,dgs),k=1..mdg-1) =0 or mul(op(k,dgs),k=mdg+1..nops(dgs)) = 0 then false; else true; end if; end if; end if; end proc:
isA183058 := proc(n) isA005384(n) and isA134808(n) ; end proc:
for n from 0 to 50000 do if isA183058(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jan 05 2011

csgpQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];PrimeQ[2n+1]&&OddQ[len]&&idn[[(len+1)/2]]==0&&Count[idn,0]==1]; Select[Prime[ Range[ 4500]],csgpQ] (* Harvey P. Dale, Jun 06 2020 *)

A005384 INTERSECT A134808.

cp's OEIS Frontend

A136098 Prime palindromic cyclops numbers.

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A182809 Cyclops-Fibonacci numbers.

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A239827 Cyclops numbers whose squares are cyclops numbers.

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A252480 Numbers whose decimal representation has at least one '0' digit in a position other than the final digit.

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Mathematica

PARI

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

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A160561 Cyclops primes with circular digits {0,6,8,9}.

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A160725 Cyclops semiprimes.

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Maple

Mathematica

A162198 Even cyclops numbers.

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