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A046851 Numbers n such that n^2 can be obtained from n by inserting internal (but not necessarily contiguous) digits. 8
0, 1, 10, 11, 95, 96, 100, 101, 105, 110, 125, 950, 960, 976, 995, 996, 1000, 1001, 1005, 1006, 1010, 1011, 1021, 1025, 1026, 1036, 1046, 1050, 1100, 1101, 1105, 1201, 1205, 1250, 1276, 1305, 1316, 1376, 1405, 9500, 9505, 9511, 9525, 9600, 9605, 9625 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Contains A038444.  In particular, the sequence is infinite. - Robert Israel, Oct 20 2016

If n is any positive term, then b_n(k) := n*10^k (k >= 0) is an infinite subsequence. - Rick L. Shepherd, Nov 01 2016

From Robert Israel's comment it follows that the subsequence of terms with no trailing zeros is also infinite (contains A000533). - Rick L. Shepherd, Nov 01 2016

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

EXAMPLE

110^2 = 12100 (insert "2" and "0" into "1_1_0").

MAPLE

IsSublist:= proc(a, b)

  local i, bp, j;

  bp:= b;

  for i from 1 to nops(a) do

    j:= ListTools:-Search(a[i], bp);

    if j = 0 then return false fi;

    bp:= bp[j+1..-1];

  od;

  true

end proc:

filter:= proc(n) local A, B;

  A:= convert(n, base, 10);

  B:= convert(n^2, base, 10);

  if not(A[1] = B[1] and A[-1] = B[-1]) then return false fi;

  if nops(A) <= 2 then return true fi;

  IsSublist(A[2..-2], B[2..-2])

end proc:

select(filter, [$0..10^4]); # Robert Israel, Oct 20 2016

MATHEMATICA

id[n_]:=IntegerDigits[n];

insQ[n_]:=First[id[n]]==First[id[n^2]]&&Last[id[n]]==Last[id[n^2]];

sort[n_]:=Flatten/@Table[Position[id[n^2], id[n][[i]]], {i, 1, Length[id[n]]}];

takeQ[n_]:=Module[{lst={First[sort[n][[1]]]}},

   Do[

    Do[

     If[Last[lst]<sort[n][[i]][[h]], AppendTo[lst, sort[n][[i]][[h]]]; Break[]],

     {h, 1, Length[sort[n][[i]]]}

     ],

    {i, 2, Length[sort[n]]}

    ];

   If[Length[lst]==Length[id[n]]&&lst==Sort[lst], True, False]

];

Select[Range[0, 9625], insQ[#]&&takeQ[#]&] (* Ivan N. Ianakiev, Oct 19 2016 *)

PROG

(Haskell)

import Data.List (isInfixOf)

a046851 n = a046851_list !! (n-1)

a046851_list = filter chi a008851_list where

   chi n = (x == y && xs `isSub` ys) where

      x:xs = show $ div n 10

      y:ys = show $ div (n^2) 10

   isSub [] ys       = True

   isSub _  []       = False

   isSub us'@(u:us) (v:vs)

         | u == v    = isSub us vs

         | otherwise = isSub us' vs

-- Reinhard Zumkeller, Jul 27 2011

CROSSREFS

Cf. A045953, A008851, A018834, A038444, A086457 (subsequence).

Sequence in context: A228381 A262229 A086457 * A045953 A136830 A153069

Adjacent sequences:  A046848 A046849 A046850 * A046852 A046853 A046854

KEYWORD

nonn,base,easy,nice

AUTHOR

David W. Wilson

STATUS

approved

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Last modified June 22 14:36 EDT 2017. Contains 288633 sequences.