A134808 -id:A134808 - cp's OEIS frontend

A153806 Strobogrammatic cyclops numbers.

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

A239589 Numbers whose squares are cyclops numbers.

Original entry on oeis.org

0, 105, 138, 145, 155, 179, 195, 205, 217, 226, 241, 243, 255, 257, 259, 274, 295, 305, 1054, 1068, 1082, 1091, 1114, 1127, 1136, 1158, 1162, 1175, 1192, 1196, 1221, 1229, 1233, 1237, 1261, 1269, 1273, 1277, 1281, 1308, 1323, 1327, 1338, 1364, 1375, 1386

Offset: 1

Colin Barker, Mar 24 2014

			138 is in the sequence because 138^2 = 19044, which is a cyclops number.

Colin Barker, Table of n, a(n) for n = 1..4000

Cf. A000290, A134808, A160711, A239587, A239588, A239590, A239591.

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, 2000, if(is_cyclops(n^2), s=concat(s, n))); s

A239587 Cubes that are cyclops numbers.

Original entry on oeis.org

0, 74088, 1520875, 1560896, 1860867, 2460375, 4330747, 4410944, 7880599, 123505992, 144703125, 172808693, 177504328, 179406144, 191102976, 194104539, 211708736, 232608375, 241804367, 264609288, 288804781, 295408296, 335702375, 338608873, 368601813, 374805361

Offset: 1

Colin Barker, Mar 24 2014

Intersection of A000578 (Cubes) and A134808 (Cyclops numbers).

Colin Barker, Table of n, a(n) for n = 1..1800

Cf. A000578, A134808, A160711, A239588, A239589, A239590, A239591.

cyclpsQ[n_]:=With[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1&&IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[ Range[ 800]^3,cyclpsQ]] (* Harvey P. Dale, Nov 05 2024 *)

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, 2000, if(is_cyclops(n^3), s=concat(s, n^3))); s

a(n) = A239590(n)^3.

A239588 Fourth powers that are cyclops numbers.

Original entry on oeis.org

0, 7890481, 9150625, 623201296, 981506241, 17363069361, 18945044881, 28813025536, 33871089681, 38167092496, 45954068161, 89526025681, 95565066496, 1421970391296, 1551160647936, 1736870953216, 3941340648961, 4771970346256, 5281980641536, 5436960129441

Offset: 1

Colin Barker, Mar 24 2014

Intersection of A000583 (Fourth powers) and A134808 (Cyclops numbers).

Colin Barker, Table of n, a(n) for n = 1..1500

Cf. A000583, A134808, A160711, A239587, A239589, A239590, A239591.

cn4Q[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn]; OddQ[ len] && idn[[(len+1)/2]]==0&&DigitCount[n,10,0]==1]; Select[Range[0,2000]^4, cn4Q] (* Harvey P. Dale, Dec 20 2015 *)

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, 2000, if(is_cyclops(n^4), s=concat(s, n^4))); s

a(n) = A239591(n)^4.

A239590 Numbers whose cubes are cyclops numbers.

Original entry on oeis.org

0, 42, 115, 116, 123, 135, 163, 164, 199, 498, 525, 557, 562, 564, 576, 579, 596, 615, 623, 642, 661, 666, 695, 697, 717, 721, 724, 748, 788, 806, 827, 886, 945, 961, 994, 2272, 2274, 2319, 2325, 2329, 2391, 2438, 2512, 2529, 2537, 2545, 2617, 2637, 2654

Offset: 1

Colin Barker, Mar 24 2014

			123 is in the sequence because 123^3 = 1860867, which is a cyclops number.

Colin Barker, Table of n, a(n) for n = 1..1500

Cf. A000578, A134808, A160711, A239587, A239588, A239589, A239591.

Join[{0},Select[Range[0,3000],OddQ[IntegerLength[#^3]]&&DigitCount[#^3,10,0]==1&&IntegerDigits[#^3][[(IntegerLength[ #^3]+ 1)/2]] == 0&]] (* Harvey P. Dale, Nov 05 2024 *)

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, 3000, if(is_cyclops(n^3), s=concat(s, n))); s

A239591 Numbers whose fourth powers are cyclops numbers.

Original entry on oeis.org

0, 53, 55, 158, 177, 363, 371, 412, 429, 442, 463, 547, 556, 1092, 1116, 1148, 1409, 1478, 1516, 1527, 1612, 1622, 1633, 1692, 1694, 1724, 1738, 1754, 3262, 3263, 3276, 3283, 3338, 3362, 3366, 3402, 3436, 3464, 3468, 3473, 3512, 3538, 3631, 3723, 3724, 3833

Offset: 1

Colin Barker, Mar 24 2014

			158 is in the sequence because 158^4 = 623201296, which is a cyclops number.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000583, A134808, A160711, A239587, A239588, A239589, A239590.

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, 5000, if(is_cyclops(n^4), s=concat(s, n))); s

A135627 Perfect numbers minus 1.

Original entry on oeis.org

5, 27, 495, 8127, 33550335, 8589869055, 137438691327, 2305843008139952127, 2658455991569831744654692615953842175, 191561942608236107294793378084303638130997321548169215

Offset: 1

Omar E. Pol, Nov 27 2007

nonn

Conjecture: every odd prime is a factor of at least one number in this sequence. - J. Lowell, Apr 17 2014

a(n) written in base 2 is A138831(n) which is also a member of A138148 (cyclops numbers with binary digits only), assuming there are no odd perfect numbers. - Omar E. Pol, Oct 24 2016 [clarified on Nov 03 2018].

G. C. Greubel, Table of n, a(n) for n = 1..15

Cf. A000396, A065091, A138148, A134808, A138831.

(# (# + 1))/2 & /@ Select[2^Range[100] - 1, PrimeQ] - 1 (* G. C. Greubel, Oct 24 2016, based on Harvey P. Dale code from A000396 *)

a(n) = A000396(n) - 1.

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

A239828 Cyclops numbers which are squares of cyclops numbers.

Original entry on oeis.org

0, 11025, 42025, 93025, 121308196, 121506529, 121903681, 122301481, 144408289, 144504441, 145106116, 145805625, 145902241, 169702729, 169806961, 171505216, 196308121, 196504324, 197205849, 197908624, 198105625, 198302724, 256608361, 256704484, 257409936

Offset: 1

Colin Barker, Mar 27 2014

Subsequence of A160711.

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

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

Cf. A134808, A160711, A239827.

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^2))); s

a(n) = A239827(n)^2.

cp's OEIS Frontend

A153806 Strobogrammatic cyclops numbers.

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A160717 Cyclops triangular numbers.

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A239589 Numbers whose squares are cyclops numbers.

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A239587 Cubes that are cyclops numbers.

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A239588 Fourth powers that are cyclops numbers.

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A239590 Numbers whose cubes are cyclops numbers.

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A239591 Numbers whose fourth powers are cyclops numbers.

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PARI

A135627 Perfect numbers minus 1.

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