A046851 Numbers n such that n^2 can be obtained from n by inserting internal (but not necessarily contiguous) digits.

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

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: A347182 A331604 A086457 * A045953 A136830 A153069

Adjacent sequences: A046848 A046849 A046850 * A046852 A046853 A046854

Keyword

nonn,base,easy,nice

Author

David W. Wilson

Status

approved