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 1-2 of 2 results.

A147553 Numbers k such that k^2 divides k.k where dot "." means concatenation.

Original entry on oeis.org

1, 143, 142857143, 142857142857143, 142857142857142857143, 142857142857142857142857143, 142857142857142857142857142857143, 142857142857142857142857142857142857143
Offset: 1

Views

Author

Farideh Firoozbakht, Dec 23 2008

Keywords

Comments

I proved that for n > 0, a(n+1) = (10^(6n-3) + 1)/7. Namely for n > 1, a(n) is of the form 142857.142857. ... .142857.143. Except for a(1), 11 divides all terms, so there is no prime p such that p^2 divides p.p. For n > 1, a(n).a(n)/(a(n)*a(n))=7.

Examples

			143*143|143.143 (143143/(143*143)=7) so 143 is in the sequence.
		

Crossrefs

Cf. A147554, A243162 (k^2 divides k.k.k).

Programs

  • Maple
    1, seq((10^(6*n-3)+1)/7, n=1..20); # Robert Israel, Sep 26 2016
  • Mathematica
    a[0]=1;a[n_]:=(10^(6n-3)+1)/7;Table[a[k],{k,0,8}]
    Do[d=Divisors[10^i+1];s=Select[d,Length[IntegerDigits[#]]==i&];If[Length[s]>0,Do[Print[s[[j]]],{j,Length[s]}]],{i,69}] (* Hans Havermann, May 31 2014 *)
    LinearRecurrence[{1000001,-1000000},{1,143,142857143},20] (* Harvey P. Dale, Apr 02 2018 *)

A243162 Numbers n such that n^2 divides n.n.n where dot "." means concatenation.

Original entry on oeis.org

1, 3, 13, 21, 37, 39, 91, 1443, 3367, 9901, 157737, 333667, 999001, 3075403, 9226209, 14287143, 33336667, 99990001, 1171182883, 1224848037, 1286294191, 1397863441, 1428557143, 1469179621, 1535254357, 1568996211, 1753536967, 1792076241, 1839599913, 1891910811
Offset: 1

Views

Author

Hans Havermann, May 31 2014

Keywords

Comments

Number of d-digit solutions for d = 1..100: 2, 5, 0, 3, 0, 3, 2, 3, 0, 39, 0, 2, 0, 106, 0, 3, 3, 2, 0, 441, 4, 14, 0, 5, 0, 15, 2, 283, 0, 23, 0, 61, 0, 24, 21, 4, 0, 22, 0, 240, 0, 34, 0, 96, 3, 30, 0, 6, 16, 281, 0, 216, 0, 22, 5, 3894, 2, 10, 0, 149, 2, 11, 0, 407, 0, 25, 0, 2136, 0, 53983, 0, 12, 1, 29, 11, 1872, 99, 20, 0, 6984, 0, 45, 0, 279, 32, 10, 5, 15928, 0, 213, 24, 791, 0, 20, 14, 44, 0, 713, 12, 89804.
Numbers n such that n divides 100^d+10^d+1, where 10^(d-1)<=n<10^d. - Robert Israel, Jan 11 2017

Examples

			21^2 divides 212121; 91^2 divides 919191; so both 21 and 91 are in the sequence.
		

Crossrefs

Cf. A147553 (n^2 divides n.n), A147554 (primes in this sequence).
Contains A074992 and A168624.

Programs

  • Maple
    Res:= {}:
    for d from 1 to 15 do
      Res:= Res union select(t -> t >= 10^(d-1) and t < 10^d,
       numtheory:-divisors(100^d+10^d+1))
    od:
    sort(convert(Res,list)); # Robert Israel, Jan 11 2017
  • Mathematica
    Do[d=Divisors[100^i+10^i+1];s=Select[d,Length[IntegerDigits[#]]==i&];If[Length[s]>0,Do[Print[s[[j]]],{j,Length[s]}]],{i,42}]
Showing 1-2 of 2 results.