cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

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

Views

Author

Omar E. Pol, Jun 08 2009

Keywords

Comments

Triangular numbers (A000217) that are also cyclops numbers (A134808).

Examples

			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

Links

Crossrefs

Cf. A000217, A134808, A134809, A138131, A138148, A153806, A160711, A160712.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Extensions

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

Views

Author

Colin Barker, Mar 24 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • PARI
    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

Views

Author

Colin Barker, Mar 24 2014

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Views

Author

Colin Barker, Mar 24 2014

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Views

Author

Colin Barker, Mar 24 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Colin Barker, Mar 24 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • PARI
    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

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

Views

Author

Omar E. Pol, Jun 08 2009

Keywords

Crossrefs

Cf. A134808, A134809, A134148, A138131, A153806, A160711, A160717.

Extensions

Edited by Omar E. Pol, Jul 04 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

Views

Author

Colin Barker, Mar 27 2014

Keywords

Comments

Subsequence of A160711.

Examples

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

Links

Crossrefs

Cf. A134808, A160711, A239827.

Programs

  • PARI
    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

Formula

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

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

Views

Author

Omar E. Pol, Jun 12 2009

Keywords

Comments

Cyclops numbers (A134808) that are also semiprimes (A001358).

Links

Crossrefs

Cf. A001358, A134808, A134809, A138131, A138148, A153806, A153807, A160711, A160712, A160717.

Programs

  • Maple
    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
  • Mathematica
    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 *)

A285845 Powers (A001597) that are also cyclops numbers (A134808).

Original entry on oeis.org

11025, 19044, 21025, 24025, 32041, 38025, 42025, 47089, 51076, 58081, 59049, 65025, 66049, 67081, 74088, 75076, 87025, 93025, 1110916, 1140624, 1170724, 1190281, 1240996, 1270129, 1290496, 1340964, 1350244, 1380625, 1420864, 1430416, 1490841, 1510441
Offset: 1

Views

Author

Colin Barker, Apr 27 2017

Keywords

Comments

The first term not in A160711 is 74088 = 42^3.
Intersection of A001597 and A134808. - Robert G. Wilson v, Apr 27 2017

Links

Crossrefs

Cf. A001597, A134808, A160711.

Programs

  • Mathematica
    Select[NestList[If[# == 1, 4, Min@ Table[(Floor[#^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ #}]] &, 1, 1400], Function[n, And[OddQ@ Length@ #, #[[ Ceiling[Length[#]/2] ]] == 0, DigitCount[n, 10, 0] == 1] &@ IntegerDigits@ n]] (* Michael De Vlieger, Apr 27 2017, after Robert G. Wilson v at A001597 *)
cyclopsQ[n_Integer, b_: 10] := Module[{digitList = IntegerDigits[n, b], len, pos0s, flag}, len = Length[digitList]; pos0s = Select[Range[len], digitList[[#]] == 0 &]; flag = OddQ[len] && (Length[pos0s] == 1) && (pos0s == {(len + 1)/2}); Return[flag]]; (* from Alonso del Arte in A134808 *) min = 0; max = 1520000; t = Union@ Flatten@ Table[n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}]; Select[t, cyclopsQ] (* Robert G. Wilson v, Apr 27 2017 *)
  • PARI
    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)}
L=List(); for(n=1, 100000, if(ispower(n) && is_cyclops(n), listput(L, n))); Vec(L)
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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