A134809 -id:A134809 - cp's OEIS frontend

A162198 Even cyclops numbers.

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.

A329737 Cyclops primes that remain prime after being "blinded".

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 503, 509, 601, 607, 701, 709, 809, 907, 11071, 11087, 11093, 12037, 12049, 12097, 13099, 14029, 14033, 14051, 14071, 14081, 14083, 14087, 15031, 15053, 15083, 16057, 16063, 16067, 16069, 16097, 17021, 17033, 17041, 17047, 17053

Offset: 1

Rodolfo Ruiz-Huidobro, Nov 20 2019

There are 14 of these primes with 3 digits and 302 with 5 digits.

			The first term, a(1), is 101 because if you remove the "cyclops' eye" it remains a prime (11) and because 101 is the 1st cyclops prime.
307 is a term because when you remove the "0" it remains a prime: 37.

Rodolfo Ruiz-Huidobro, Table of n, a(n) for n = 1..3194

Intersection of A256186 and A134809.

a:=[]; f:=func; g:=func; for n in [1..20000] do if f(n) and IsPrime(g(n)) then Append(~a,n); end if; end for; a; // Marius A. Burtea, Nov 20 2019

A183059 Cyclops primes p such that 2p+1 is also a Cyclops prime.

Original entry on oeis.org

12011, 12041, 13049, 18041, 21011, 22013, 28019, 29021, 29033, 31019, 33023, 37013, 37049, 38039, 42023, 43013, 48029, 1110269, 1120349, 1120481, 1130273, 1130429, 1140143, 1140311, 1140341, 1140383, 1140413, 1140449, 1160129, 1160213, 1160429, 1170173, 1170329, 1170443

Offset: 1

Omar E. Pol, Dec 25 2010

Primes p such that both p and 2p+1 are Cyclops primes A134809.

By definition all terms are also Sophie Germain primes A005384.

			a(1) = 12011 is in the sequence because 12011 is a Cyclops prime A134809 and 2*12011+1 = 24023 is also a Cyclops prime.

Cf. A005384, A134808, A134809, A183058.

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:
isA134809 := proc(n) isprime(n) and isA134808(n) ; end proc:
isA183059 := proc(n) isA134809(n) and isA134809(2*n+1) ; end proc:
for n from 0 to 1200000 do if isA183059(n) then printf("%d,",n); end if; end do: # R. J. Mathar, Jan 05 2011

A330414 Cyclops primes that become a cube when the middle "0" is removed.

Original entry on oeis.org

68059, 1170649, 4560533, 7530571, 136501919, 158103251, 173703979, 212503933, 226605187, 356101289, 362604691, 382702753, 439806977, 518905117, 811802737, 954403993, 19484041249, 19956016979, 22635071297, 24658046551, 27263097773, 34635012697, 35326042667, 37166072149, 39668022287, 41499095543, 44839062449

Offset: 1

Rodolfo Ruiz-Huidobro, Dec 14 2019

			a(1) = 68059 because 6859 = 19^3 is the first cube that results from the removal of the 0 digit from a cyclops prime.
136501919 is a term because 13651919 is 239^3.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 60 terms from Rodolfo Ruiz-Huidobro)

Cf. A329737, A134809, A016755.

count:= 0: Res:= NULL:
for d from 2 to 6 do
  for n from ceil(10^((2*d-1)/3)) to floor((10^(2*d)-1)^(1/3)) do
    L:=convert(n^3,base,10);
    if member(0,L) then next fi;
    a:= n^3 mod 10^d;
    p:= 10*(n^3-a)+a;
    if isprime(p) then
      count:= count+1; Res:= Res, p;
    fi
od od:
Res; # Robert Israel, Dec 24 2019

seq(n)={my(i=0, L=List()); while(#Lt==0,v), my(m=fromdigits(concat([v[1..k], 0, v[k+1..#v]]))); if(isprime(m), listput(L,m)))); Vec(L)} \\ Andrew Howroyd, Dec 20 2019

A162199 Odd cyclops numbers.

Original entry on oeis.org

101, 103, 105, 107, 109, 201, 203, 205, 207, 209, 301, 303, 305, 307, 309, 401, 403, 405, 407, 409, 501, 503, 505, 507, 509, 601, 603, 605, 607, 609, 701, 703, 705, 707, 709, 801, 803, 805, 807, 809, 901, 903, 905, 907, 909, 11011, 11013, 11015, 11017

Offset: 1

Omar E. Pol, Jul 04 2009

Cf. A134808, A134809, A162198.

isA134808 := proc(n) local dgs,ndgs,d ; dgs := convert(n,base,10) ; ndgs := nops(dgs) ; if ndgs mod 2 = 0 then false; else for d from 1 to ndgs do if d = (ndgs+1)/2 and op(d,dgs) <> 0 then RETURN(false); elif d <> (ndgs+1)/2 and op(d,dgs) = 0 then RETURN(false); fi; od: true; fi; end: for n from 1 to 11100 by 2 do if isA134808(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009

More terms from R. J. Mathar, Jul 13 2009

A183056 Cyclops partition numbers.

Original entry on oeis.org

101, 26015, 483502844, 27517052599, 2814570987591, 269232701252579, 12269218019229465, 477535459708164115593, 55271949286085137715955, 98175979536033971312388, 28848173767368633057992125893483779

Offset: 1

Omar E. Pol, Dec 21 2010

			a(1) = 101 is in the sequence because 101 is a partition number A000041 and it is also a cyclops number A134808.

Cf. A000041, A134808, A134809, A160717.

cnQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];DigitCount[ n,10,0]==1&&OddQ[len]&&idn[[(len+1)/2]]==0]; Select[PartitionsP[ Range[ 2000]],cnQ] (* Harvey P. Dale, Apr 10 2019 *)

Intersection of A000041 and A134808.

a(11) from Alois P. Heinz, Dec 23 2010

A183057 Cyclops emirps.

Original entry on oeis.org

107, 701, 709, 907, 11057, 11071, 11083, 12071, 12073, 13043, 14029, 14057, 14071, 14081, 14087, 15013, 15053, 15091, 16063, 16073, 17011, 17021, 17033, 17041, 17047, 18013, 18041, 18077, 18089, 19013, 19037, 19051, 31033, 31051, 31063, 31069, 31081, 31091, 32077, 32099

Offset: 1

Omar E. Pol, Dec 21 2010

Intersection of emirps A006567 and cyclops numbers A134808.

The smallest cyclops emirp 107 was mentioned by Patrick Capelle in Prime Curios! (see link).

			a(1) = 107 is in the sequence because 107 is an emirp A006567 and it is also a cyclops number A134808.

G. L. Honaker, Jr. and C. K. Caldwell, Prime Curios! 107

Cf. A006567, A134808, A134809, A138148, A160712, A160725.

A006567 INTERSECT A134808.

A285767 Cyclops octagonal numbers: a(n) = n(3n-2) with one "zero" digit in the middle.

Original entry on oeis.org

0, 408, 11041, 18096, 22016, 23056, 28033, 38081, 56033, 61061, 1140833, 1170625, 1250656, 1410416, 1460216, 1540833, 2120161, 2130261, 2140385, 2150533, 2310896, 2390561, 2460696, 2520833, 2570576, 2780181, 2920533, 3230256, 3280256, 3490565, 3660865, 3680776

Offset: 1

K. D. Bajpai, Apr 25 2017

The n-th octagonal number x(n) = n*(3*n - 2).
Subset of A000567.
All the terms have the number of digits odd with only one "zero" digit in the middle.

			For n = 12; x(12) = 12*(3*12 - 2) = 408 that is 12th octagonal number with one zero digit in the middle, hence appears in the sequence.
For n = 61; x(61) = 61*(3*61 - 2) = 11041 that is 61st octagonal number with one zero digit in the middle, hence appears in the sequence.

Intersection of A000567 and A134808.

Cf. A000567, A134808, A134809, A160717.

iscyclops:= proc(n) local L,t;
t:= ilog10(n);
if t::odd then return false fi;
L:= convert(n,base,10);
L[1+t/2] = 0 and numboccur(0,L) = 1
end proc:
iscyclops(0):= true:
select(iscyclops, [seq(n*(3*n-2),n=0..1000)]);

Select[Table[n (3 n - 2), {n, 0, 1110}], And[OddQ@ Length@ #, Count[#, 0] == 1, Take[#, {Ceiling[Length[#]/2]}] == {0}] &@ IntegerDigits@ # &] (* Michael De Vlieger, Apr 26 2017 *)

cp's OEIS Frontend

A162198 Even cyclops numbers.

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A182811 Cyclops-Lucas numbers.

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A183058 Cyclops Sophie-Germain primes.

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A329737 Cyclops primes that remain prime after being "blinded".

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Magma

A183059 Cyclops primes p such that 2p+1 is also a Cyclops prime.

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Maple

A330414 Cyclops primes that become a cube when the middle "0" is removed.

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PARI

A162199 Odd cyclops numbers.

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A183056 Cyclops partition numbers.

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A285767 Cyclops octagonal numbers: a(n) = n(3n-2) with one "zero" digit in the middle.