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.

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A137146 Numbers k such that k and k^2 use only the digits 5, 6, 7 and 8.

Original entry on oeis.org

76, 766, 7666, 76666, 766666, 7666666, 76666666, 766666666, 7666666666, 76666666666, 766666666666, 7666666666666, 76666666666666, 766666666666666, 7666666666666666, 76666666666666666, 766666666666666666, 7666666666666666666, 76666666666666666666, 766666666666666666666
Offset: 1

Views

Author

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Keywords

Comments

Generated with DrScheme.
The first digit of each term is either 7 or 8 and the last digit is 6. - Chai Wah Wu, May 25 2021

Examples

			766666666666666^2 = 587777777777776755555555555556.

Links

Crossrefs

Cf. A000290 (the squares); A136808, A136809, ..., A137147 for other digit combinations.
Cf. A058469 - A058472 and A058411, ..., A058474 for other digit combinations.
Cf. A277959, A277960, A277961, A295005, ..., A295009 (squares with largest digit = 2, 3, 4, 5, ..., 9).

Programs

  • Python
    from itertools import product
A137146_list = [n for n in (int(''.join(d)) for l in range(1,6) for d in product('5678',repeat=l)) if set(str(n**2)) <= set('5678')] # Chai Wah Wu, May 25 2021

Extensions

a(15)-a(20) from Pontus von Brömssen, Apr 12 2024

A137144 Numbers k such that k and k^2 use only the digits 4, 6, 7 and 8.

Original entry on oeis.org

8, 88, 8874, 68474, 86478
Offset: 1

Views

Author

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Keywords

Comments

Generated with DrScheme.
No further terms up to and including 1000000. - Harvey P. Dale, Dec 03 2010
No further terms <= 10^40. - Michael S. Branicky, Feb 12 2024
From Pontus von Brömssen, May 01 2024: (Start)
a(6) > 6*10^46 (if it exists).
If k = x*10^m is a term where 1 < x < 10 and k is not 88 or 8874, then 20/3 < x < 8.7674847468864688448864887688468686674647846475.
(End)

Examples

			86478^2 = 7478444484.

Links

Crossrefs

Cf. A077439, A136809, A136813, A136985, A137028, A137105, A137147.

Programs

  • Mathematica
    clearQ[n_]:=Module[{dc=DigitCount[n]},dc[[1]]==dc[[2]]==dc[[3]]==dc[[5]]==dc[[9]]==dc[[10]]==0]
Select[Range[1000000],clearQ[#]&&clearQ[#^2]&] (* Harvey P. Dale, Dec 03 2010 *)

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35, 38, 40, 41, 45, 46, 50, 55, 56, 60, 63, 64, 65, 66, 68, 70, 74, 75, 76, 77, 80, 81, 83, 85, 88, 90, 91, 95, 96, 97, 99, 100, 101, 102, 105, 109, 110
Offset: 1

Views

Author

Jovan Radenkovicc, Nov 15 2024

Keywords

Comments

Problem: Is there a real constant c such that a(n) < n^c for all positive integers n?
All of A136808, A136809, A136816, ..., A137079 are subsequences. Many if not most terms of A058411, A058413, ... ("tridigital solutions") are also in this sequence; see also Hisanori Mishima's web page for some nontrivial solutions. - M. F. Hasler, Feb 02 2025

Examples

			816 is in the sequence since 816^2 = 665856 and both together use at most 4 distinct digits.
149 is not in the sequence since 149^2 = 22201 and both together use 5 distinct digits.

Links

Crossrefs

Supersequence of A136808, A136809, A136816, A136822.
Cf. A043537, A053061, A055436, A154155.

Programs

  • Magma
    [n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];
  • Mathematica
    Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)
  • PARI
    isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024
  • PARI
    is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)),s)<5 \\ Charles R Greathouse IV, Jan 30 2025
  • PARI
    is1(n)=#setunion(Set(digits(n^2)),Set(digits(n)))<5
ok(m)=my(d=concat(apply(k->digits(lift(k)), [m,m^2]))
test(d)=my(v=List(),D=10^d); for(n=0,D-1, if(ok(Mod(n,D)), listput(v,n))); Vec(v)
res=test(8); \\ build a list of residues mod 10^8
D=diff(concat(res,res[1]+10^8)); #D
u=List(); for(n=0,10^7, if(is1(n) && !setsearch(n,res), listput(u,n))); \\ build exceptions
setminus(select(is1,[0..n]),list(n))
list(lim)=my(v=List(u)); forstep(n=0,lim,D, if(is1(n), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 30 2025
  • Python
    def ok(n): return len(set(str(n)+str(n**2))) <= 4
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Nov 18 2024

Formula

A043537(A053061(a(n))) <= 4.
Trivially, a(n) >> n^1.66... where the exponent is log(10)/log(4) (A154155). - Charles R Greathouse IV, Jan 30 2025

A137078 Numbers k such that k and k^2 use only the digits 2, 3, 4 and 9.

Original entry on oeis.org

2, 3, 4943, 499423, 49993443, 4999932923, 499999999293429243923, 499999999999293429243923
Offset: 1

Views

Author

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Keywords

Comments

Generated with DrScheme.

Examples

			4999932923^2 = 24999329234499323929.

Links

Crossrefs

Cf. A136809.

Programs

  • Maple
    g:= n -> convert(convert(n,base,10),set) subset {2,3,4,9}:
Res:= 2,3:
extend:= proc(n,d) local B; global Res;
B:= {seq(x*10^d+n,x=[2,3,4,9])};
Res:= Res, op(select(t -> g(t^2), B));
op(select(t -> g(t^2 mod 10^(d+1)), B))
end proc:
Agenda:= {2,3,4,9}:
for d from 1 to 25 do Agenda:= map(extend,Agenda,d) od:
sort([Res]); # Robert Israel, Oct 29 2018

Extensions

a(7)-a(8) from Andrew Howroyd, Oct 24 2018
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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