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 1-8 of 8 results.

A235717 Squares which have one or more occurrences of exactly two different digits.

Original entry on oeis.org

16, 25, 36, 49, 64, 81, 100, 121, 144, 225, 400, 441, 484, 676, 900, 1444, 7744, 10000, 11881, 29929, 40000, 44944, 55225, 69696, 90000, 1000000, 4000000, 9000000, 9696996, 100000000, 400000000, 900000000, 6661661161, 10000000000, 40000000000, 90000000000
Offset: 1

Views

Author

Colin Barker, Jan 15 2014

Keywords

Comments

The first term having a repeated digit is 100.
This sequence is the same as A018885, except that A018885 has four additional leading terms.

Examples

			69696 is in the sequence because 69696 = 264^2 and 69696 contains exactly two different digits: 6 and 9.

Crossrefs

Cf. A235718-A235724, A225218.

Programs

  • PARI
    s=[]; for(n=1,10000, if(#vecsort(eval(Vec(Str(n^2))),,8)==2, s=concat(s, n^2))); s

Formula

a(n) = A016069(n)^2.

A054031 Numbers whose square contains exactly 3 distinct digits.

Original entry on oeis.org

13, 14, 16, 17, 18, 19, 23, 24, 25, 27, 28, 29, 31, 34, 35, 39, 40, 41, 45, 46, 47, 50, 56, 58, 60, 62, 63, 65, 67, 68, 70, 75, 76, 77, 80, 81, 83, 85, 90, 91, 92, 94, 97, 101, 102, 107, 108, 110, 111, 119, 120, 121, 122, 129, 131, 141, 149, 150, 162, 165
Offset: 1

Views

Author

Asher Auel, Feb 29 2000

Keywords

Links

Crossrefs

Cf. A235718, A016069, A054032, A054033, A054034, A054035, A054036, A054037, A054038, A054039.

Programs

  • Maple
    f := []; for i from 0 to 200 do if nops({op(convert(i^2,base,10))})=3 then f := [op(f),i] fi; od; f;
  • Mathematica
    t = {}; n = -1; While[Length[t] < 50, n++; If[Length[Union[IntegerDigits[n^2]]] == 3, AppendTo[t, n]]] (* T. D. Noe, Apr 26 2013 *)
Select[Range[200],Length[Union[IntegerDigits[#^2]]]==3&] (* Harvey P. Dale, Aug 17 2014 *)
  • PARI
    is(n)=#Set(digits(n^2))==3 \\ Charles R Greathouse IV, Feb 11 2017

Formula

A235718(n) = a(n)^2. - Giovanni Resta, Apr 28 2017

A235719 Squares which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1024, 1089, 1296, 1369, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4761, 5041, 5184, 5329, 5476, 6084, 6241, 6724, 7056, 7396, 7569, 7921, 8649, 9025, 9216, 9604, 9801, 10609, 10816, 11025, 11236, 12544, 12996
Offset: 1

Views

Author

Colin Barker, Jan 15 2014

Keywords

Comments

The first term having a repeated digit is 10609.

Examples

			5329 is in the sequence because 5329 = 73^2 and 5329 contains exactly four different digits: 2, 3, 5 and 9.

Links

Crossrefs

Cf. A235717, A235718, A235720-A235724, A225218.

Programs

  • Mathematica
    Select[Range[150]^2,Length[Union[IntegerDigits[#]]]==4&] (* Harvey P. Dale, May 03 2018 *)
  • PARI
    s=[]; for(n=1, 300, if(#vecsort(eval(Vec(Str(n^2))),,8)==4, s=concat(s, n^2))); s

Formula

a(n) = A054032(n)^2.

A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

Original entry on oeis.org

0, 0, 648, 3888, 16200, 58320, 195048, 625968, 1960200, 6045840, 18468648, 56068848, 169533000, 511252560, 1539065448, 4627812528, 13904670600, 41756478480, 125354369448, 376232977008, 1129038669000, 3387795483600, 10164745404648, 30496954122288, 91496298184200
Offset: 1

Views

Author

Stefano Spezia, Aug 22 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A031962.

Examples

			a(1) = a(2) = 0 since the positive integers must have at least three digits;
a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;
a(4) = 3888 since #[9999] - #[999] - #(1111*[9]) - A335843(4) - #{xywz in N | x,y,w,z are four different digits with x != 0} = 9999 - 999 - 9 - 567 - 9*9*8*7 = 3888;
...

Links

Crossrefs

Cf. A000225, A000244, A000392, A031962, A048993, A335843, A337127.
Cf. A055642, A180599, A235155, A235691, A235718.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6},{0,0,648},26]
  • PARI
    concat([0,0],Vec(648*x^3/(1-6*x+11*x^2-6*x^3)+O(x^26)))

Formula

O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).
E.g.f.: 108*(exp(x) - 1)^3.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).
a(n) = 324*(3^(n-1) - 2^n + 1).
a(n) ~ 108 * 3^n.
a(n) = 324*(A000244(n-1) - A000225(n)).
a(n) = A337127(n, 3).

A030295 Cubes with at most three distinct digits.

Original entry on oeis.org

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 2744, 3375, 8000, 27000, 46656, 64000, 238328, 343000, 778688, 1000000, 1030301, 1331000, 5177717, 7077888, 8000000, 9393931, 27000000, 64000000, 343000000, 700227072
Offset: 1

Views

Author

Patrick De Geest

Keywords

Links

Crossrefs

Cf. A030293 (subsequence), A030294, A235718.

Programs

  • Mathematica
    Select[Range[900]^3, Length@ Union@ IntegerDigits[#] <= 3 &] (* Michael De Vlieger, Feb 10 2020 *)
  • PARI
    disdigs(n,nd)={my(v=vector(10),d=digits(n^3));for(j=1,#d,v[d[j]+1]=1);if(vecsum(v)<=nd,n^3,0)};
print1(0,", ");for(k=1,1000,if(j=disdigs(k,3),print1(j,", "))) \\ Hugo Pfoertner, Feb 10 2020

Formula

a(n) = A030294(n)^3. - Peter Munn, Feb 02 2020

A378492 Squares where larger digits have larger multiplicity.

Original entry on oeis.org

0, 1, 4, 9, 144, 441, 1444, 29929, 55225, 166464, 255025, 299209, 633616, 646416, 767376, 4999696, 9696996, 34433424, 228281881, 414041104, 414488881, 424442404, 536663556, 969699600, 1649496996, 1929229929, 2636206336, 2666999449, 2929299129, 2996029696, 4664343616
Offset: 1

Views

Author

Erich Friedman, Nov 28 2024

Keywords

Links

Crossrefs

Cf. A000290, A018884, A052046, A235718, A378498.

Programs

  • Maple
    filter:= proc(n) local L,S;
   L:= convert(n,base,10);
   S:= Statistics:-Tally(L,output=list);
   S:= sort(S, (a,b) -> lhs(a) < lhs(b));
   andmap(t -> rhs(S[t])Robert Israel, Nov 29 2024
  • Mathematica
    increasingQ[L_]:=Min[Rest[(L-RotateRight[L])]]>0;
sortedQ[n_]:=increasingQ[Sort[Tally[IntegerDigits[n]]][[All,2]]]
Select[Range[575000000]^2,sortedQ]

A378498 Squares where larger digits have smaller multiplicity.

Original entry on oeis.org

1, 4, 9, 100, 121, 225, 400, 484, 676, 900, 10000, 11881, 40000, 44944, 69696, 90000, 111556, 202500, 220900, 225625, 232324, 261121, 265225, 300304, 442225, 444889, 695556, 1000000, 1002001, 1020100, 1210000, 2250000, 2295225, 4000000, 4008004, 4080400, 4840000, 5112121, 6760000, 8008900, 9000000
Offset: 1

Views

Author

Erich Friedman, Nov 28 2024

Keywords

Comments

Conjecture: a(n) ≍ n^2. - Charles R Greathouse IV, Nov 29 2024

Links

Crossrefs

Cf. A000290, A018884, A052046, A235718, A378492.

Programs

  • Mathematica
    decreasingQ[L_]:=Max[Rest[(L-RotateRight[L])]]<0;
sortedQ[n_]:=decreasingQ[Sort[Tally[IntegerDigits[n]]][[All,2]]];
Select[Range[10000]^2, sortedQ]
  • PARI
    has(n)=my(d=matreduce(digits(n))[,2]); for(i=2,#d, if(d[i]>=d[i-1], return(0))); 1
list(lim)=my(v=List()); for(n=1,sqrtint(lim\1), if(has(n^2), listput(v,n^2))); Vec(v) \\ Charles R Greathouse IV, Nov 29 2024

Formula

n^2 << a(n) << 1.001^n. - Charles R Greathouse IV, Nov 29 2024

A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

Original entry on oeis.org

0, 0, 0, 4536, 45360, 294840, 1587600, 7715736, 35244720, 154700280, 661122000, 2773768536, 11487556080, 47136955320, 192126589200, 779279814936, 3149513947440, 12695388483960, 51073849285200, 205172877726936, 823325141746800, 3301203837670200, 13228529919066000
Offset: 1

Views

Author

Stefano Spezia, Sep 26 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A031969.

Examples

			a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;
a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;
a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;
...

Links

Crossrefs

Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.
Cf. A055642, A180599, A235155, A235691, A235718.

Programs

  • Mathematica
    LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]
  • PARI
    concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

Formula

O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).
E.g.f.: 189*(exp(x) - 1)^4.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.
a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).
a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).
a(n) ~ 189 * 4^n.
a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).
a(n) = A337127(n, 4).
Showing 1-8 of 8 results.

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