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Numbers whose square is a permutational number A134640.
3

%I #17 Sep 04 2020 21:23:17

%S 0,1,15,50,85,195,327,561,789,867,1323,1764,2450,2751,2858,2878,3213,

%T 3418,3538,4834,4846,5062,5342,5770,6286,7814,8574,8634,9722,10254,

%U 10610,10614,11522,11702,11826,12363,12543,13490,14246,14502,14538,14676,14818,14902,15186,15434,15681,15874,15963

%N Numbers whose square is a permutational number A134640.

%H Robert Israel, <a href="/A134742/b134742.txt">Table of n, a(n) for n = 1..261</a>

%F a(n) = sqrt(A134741(n)).

%p N:= 10^5: # for terms <= N

%p extend:= proc(x,N,S,b,k)

%p local i,R;

%p R:= NULL;

%p for i in S while x + i*b^k <= N^2 do

%p if k = 0 then

%p if issqr(x+i*b^k) then R:= R, sqrt(x+i*b^k) fi

%p else

%p R:= R, procname(x+i*b^k,N,subs(i=NULL,S),b,k-1)

%p fi

%p od;

%p R

%p end proc:

%p f:= (b,N) -> extend(0,N,[$0..(b-1)],b,b-1):

%p R:= 0:

%p for b from 2 while b^(b-2) < N^2 do

%p R:= R, f(b,N);

%p od:

%p sort([R]); # _Robert Israel_, Sep 04 2020

%t a = {}; b = {}; Do[AppendTo[b, n]; w =Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; If[IntegerQ[j^(1/2)], AppendTo[a, j]], {m, 1, Length[w]}], {n, 0, 7}]; Sqrt[a]

%Y Cf. A134640, A134641, A134642, A134643, A134644, A023811, A134741.

%K nonn

%O 1,3

%A _Artur Jasinski_, Nov 07 2007

%E Corrected and more terms from _Robert Israel_, Sep 04 2020