A134808 -id:A134808 - cp's OEIS frontend

A274765 Cyclops numbers with circular digits {0,6,8,9}.

0, 606, 608, 609, 806, 808, 809, 906, 908, 909, 66066, 66068, 66069, 66086, 66088, 66089, 66096, 66098, 66099, 68066, 68068, 68069, 68086, 68088, 68089, 68096, 68098, 68099, 69066, 69068, 69069, 69086, 69088, 69089, 69096, 69098, 69099, 86066, 86068, 86069, 86086, 86088, 86089, 86096, 86098, 86099

Offset: 1

Kenny Lau, Jul 05 2016

Intersection of A001743 and A134808.

			86069 is a member because it is cyclops (A134808) and each digit contains at least one loop (A001743).

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

Cf. A001633, A001742, A134808, A160561.

cncdQ[n_]:=Module[{idn=IntegerDigits[n]},OddQ[Length[idn]]&&Count[idn,0] == 1&&idn[[(Length[idn]+1)/2]]==0&&SubsetQ[{0,6,8,9},idn]]; Select[ Range[ 0,90000],cncdQ] (* Harvey P. Dale, Jan 06 2022 *)

is_a001633(n) = #Str(n)%2==1
is_a001743(n) = #setintersect([1, 2, 3, 4, 5, 7], Set(digits(n)))==0
is_a134808(n) = if(n==0, return(1), if(n < 10, return(0), my(d=digits(n), x=1, y=#d); while(x < #d, if(d[x]==0, break); x++); while(y > 1, if(d[y]==0, break); y--); if(x==y && x==ceil(#Str(n)/2), return(1), return(0))))
is(n) = is_a001633(n) && is_a001743(n) && is_a134808(n) \\ Felix Fröhlich, Jul 05 2016

import sys
f = open('b274765.txt', 'w')
i = 1
n = 0
a = [""]
while True:
    for x in a:
        for y in a:
            f.write(str(i)+" "+x+"0"+y+"\n")
            i += 1
            if i>20000:
                f.close()
                sys.exit()
    a = sum([[x+"6", x+"8", x+"9"] for x in a], [])
# Kenny Lau, Jul 05 2016

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

A274612 Numbers not divisible by 10 with at least one zero but no two adjacent 0's among its decimal digits.

Original entry on oeis.org

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, 607, 608, 609, 701, 702, 703, 704, 705, 706, 707, 708, 709, 801, 802, 803, 804, 805, 806, 807, 808, 809, 901, 902, 903, 904, 905, 906, 907, 908, 909, 1011

Offset: 1

José de Jesús Camacho Medina, Jun 30 2016

			101 is a term because 101 = 101  + (1^0 * 0^1);
1010 is a term because 1010 = 1010 + (1^0 * 0^1 * 1^0 ).

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

Cf. A134808, A005188.

filter:= proc(n) local L;
  L:= [ListTools:-SearchAll(0,convert(n,base,10))];
  nops(L) >= 1 and L[1]<>1 and not has(L[2..-1]-L[1..-2],1)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 24 2016

AA = Table[(Product[(Mod[(Floor[f/10^n]), 10])^(Mod[(Floor[f/10^(n - 1)]), 10]), {n, 1, Floor[Log[10, f]]}]), {f, 1, 1200}]
BB = Table[n, {n, 1, 1200}]
Position[(AA + BB) - BB, 0]

is(n)=n%10 && vecmin(digits(n))==0 && vecmin(digits(n,100)) && vecmin(digits(n\10,100)) \\ Charles R Greathouse IV, Jul 18 2016

A279260 Numbers which are cyclops palindromic in their binary reflected Gray code representation.

Original entry on oeis.org

0, 6, 18, 90, 330, 1386, 5418, 21930, 87210, 349866, 1397418, 5593770, 22366890, 89483946, 357903018, 1431677610, 5726579370, 22906579626, 91625794218, 366504225450, 1466014804650, 5864063412906, 23456245263018, 93824997829290, 375299957762730, 1501199898159786, 6004799458421418

Offset: 0

Indranil Ghosh, Jan 17 2017

Cyclops palindromic numbers in base 2 are numbers with middle bit 0, having equal number of 1's on both side of the 0. There is a single 0 bit in the middle and the total number of bits is odd. The middle '0' represents the eye of a cyclops.

a(n) mod 6 = 0.

			90 is in the sequence because the binary reflected Gray code representation of 90 is '1110111' which is a cyclops palindromic binary number.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Indranil Ghosh, Proof of 6|{(-2*(1+((-2)^n)-(2^(2*n+1))))/3}
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, (2015)

Cf. A014550, A129868, A134808, A138148.

Partial sums of A071930.

a(n)=(-2*(1+((-2)^n)-(2^(2*n+1))))/3 \\ Charles R Greathouse IV, Jun 29 2018

def a(n):
    return (-2*(1+((-2)**n)-(2**(2*n+1))))/3

a(n) = (-2*(1+((-2)^n)-(2^(2*n+1))))/3.

A285986 Numbers that are never cyclops for any base b > 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 30, 31, 32, 36, 42, 43, 44, 45, 46, 47, 48, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 80, 81, 90, 91, 92, 93, 94, 95, 96, 97, 98, 114, 117, 118, 120, 121, 136, 137, 138, 141, 144, 156, 157, 158, 159

Offset: 1

Rémy Sigrist, Apr 30 2017

A134808 gives the definition of cyclops numbers for base 10; we can naturally generalize this notion for any base b > 1.

			The following table indicates why 42 is not cyclops for any base b > 1:
b    42 in base b    Reason
-    ------------    ------
2    1,0,1,0,1,0     Even number of digits
3    1,1,2,0         Even number of digits
4    2,2,2           No middle 0
5    1,3,2           No middle 0
6    1,1,0           No middle 0
7    6,0             Even number of digits
...  X,X             Even number of digits
42   1,0             Even number of digits
>42  42              No middle 0
Hence 42 appears in the sequence.
The number 51 is cyclops for bases 4 (303), 5 (201) and 7 (102); hence 51 does not appear in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A134808.

is(n) = if (n==0, return (0)); my (base=2); while (1, my (d=digits(n, base)); if (#d<3, return (1)); if (#d%2==1 && d[(#d+1)/2]==0 && sum(i=1,#d,1-sign(d[i]))==1, return (0)); base++)

A285987 a(n)=least number that is cyclops for exactly n distinct bases b > 1.

Original entry on oeis.org

1, 5, 19, 51, 197, 901, 2038, 5839, 13476, 18517, 36993, 60544, 60557, 60555, 60553, 705691, 705758, 2528254, 4762881, 5645512, 5645535, 20250250, 20250249, 26733097, 66978018, 66978001, 66978016, 66978065, 66978193, 133956385, 187142781, 589907497, 1018886425

Offset: 0

Rémy Sigrist, Apr 30 2017

A134808 gives the definition of cyclops numbers for base 10; we can naturally generalize this notion for any base b > 1.

This sequence is not strictly increasing; for example: a(12) = 60557 > a(13) = 60555 > a(14) = 60553.

Rémy Sigrist, Table of n, a(n) for n = 0..35
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, C program for A285987

Cf. A134808.

A328256 Amicable pairs of cyclops numbers.

Original entry on oeis.org

1280565, 1340235, 71526069316, 75257076284, 1453520375775, 1561230417825, 1568650892445, 1995690781539, 2714480497936, 2854320218864, 5776910172896, 5864460215584, 5818350517628, 6516910297732, 6111770382135, 6139270339785, 9176850735616, 9194440569344, 114552504952875, 123277906567125

Offset: 1

Omar E. Pol, Oct 09 2019

Amicable pairs (A259180) where both members are cyclops numbers (A134808).
Is this sequence finite? What is the largest known pair?
Up to 2^64 there are 439 pairs, the largest of them is {9591988390446931328, 9596251990981497472} (Eldar).

			{1280565, 1340235} is the first amicable pair where both members are cyclops numbers, so a(1) = 1280565 and a(2) = 1340235.

Amiram Eldar, Table of n, a(n) for n = 1..878
OEIS Wiki, Cyclops Numbers

Subsequence of A259180.

Cf. A002025, A002046, A134808, A262622, A262623.

More terms from Amiram Eldar, Oct 09 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.

cp's OEIS Frontend

A274765 Cyclops numbers with circular digits {0,6,8,9}.

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

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A274612 Numbers not divisible by 10 with at least one zero but no two adjacent 0's among its decimal digits.

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A279260 Numbers which are cyclops palindromic in their binary reflected Gray code representation.

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A285986 Numbers that are never cyclops for any base b > 1.

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A285987 a(n)=least number that is cyclops for exactly n distinct bases b > 1.

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A328256 Amicable pairs of cyclops numbers.

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A162199 Odd cyclops numbers.

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A183056 Cyclops partition numbers.