A279410 - OEIS

%I #13 Dec 23 2024 14:53:45

%S 35,38,46,65,76,83,85,318,335,348,359,380,383,393,400,415,419,432,436,

%T 441,457,469,500,511,526,527,585,586,599,600,611,620,636,648,654,665,

%U 688,692,696,700,711,718,728,752,755,771,781,786,793,800,809,811,826,828,832,834,836,838,857,866,880,900,908,911,922,928,944,951,958,995

%N Numbers whose squares have identical middle digits in base 10

%C The sequence of squares starts: 1225, 1444, 2116, 4225, 5776, 6889, 7225, 101124, 112225, 121104, 128881, 144400, ...

%C By definition the sequence only contains numbers whose square has an even number of digits in base 10.

%C The sequence of middle digits starts: 2, 4, 1, 2, 7, 8, 2, 1, 2, 1, 8, 4, 6, 4, 0, ...

%H Robert Israel, <a href="/A279410/b279410.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-December/017132.html">Mail by Lars Blomberg to the Seqfan Mailing list Dec 12 2016</a>

%e 46 is in this sequence because its square, 2116, has its two middle digits equal to 1.

%p a:= proc(n) option remember; local k, kk, t;

%p       for k from 1+`if`(n=1, 0, a(n-1)) do kk:=k^2;

%p         t:= length(kk);

%p         if t::even and irem(parse(substring(""||kk,

%p            t/2..t/2+1)), 11)=0 then return k fi

%p       od

%p     end:

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Dec 22 2016

%t TakeEvenCenter[k_List] :=

%t If[EvenQ[Length[k]], k[[{Length[k]/2, Length[k]/2 + 1}]], {}]; Module[{rz},

%t Select[Range[

%t    1000], (rz = TakeEvenCenter[IntegerDigits[#^2, 10]];

%t     Length[rz] == 2 && Equal @@ rz) &]]

%K nonn,base

%O 1,1

%A _Olivier Gérard_, Dec 12 2016