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A248051 Numbers whose cubes become squares if some digit is prepended, inserted or appended.

%I A248051 #78 Mar 03 2019 01:54:26
%S A248051 1,2,5,6,10,25,30,40,41,60,84,90,96,100,121,129,160,169,200,201,250,
%T A248051 266,360,400,490,500,600,640,724,810,1000,1025,1210,1440,1690,1960,
%U A248051 2250,2500,2560,2890,3000,3240,3604,3610,4000,4100,4410,4840,5216,5290,5760
%N A248051 Numbers whose cubes become squares if some digit is prepended, inserted or appended.
%C A248051 No leading zeros allowed.
%C A248051 Number of terms <= 10^k for k = 0, 1, 2, ...: 1, 5, 14, 31, 64, 144, 373, ..., . _Robert G. Wilson v_, Dec 27 2016
%H A248051 Robert G. Wilson v, <a href="/A248051/b248051.txt">Table of n, a(n) for n = 1..373</a> (terms 1..100 from Paolo P. Lava, terms 101..144 from Davin Park)
%e A248051 If n = 1 then n^3 = 1 and if we append a 6 we have sqrt(16) = 4.
%e A248051 If n = 2 then n^3 = 8 and if we append a 1 we have sqrt(81) = 9.
%e A248051 If n = 5 then n^3 = 125 and if we insert a 2 we get sqrt(1225) = 35.
%e A248051 Again, if n = 25 then n^3 = 15625 and we have sqrt(105625) = 325 or sqrt(156025) = 395.
%p A248051 with(numtheory): P:=proc(q) local a,b,j,k,n,ok;
%p A248051 for n from 1 to q do a:=n^3; b:=ilog10(a)+1; ok:=1;
%p A248051 for k from 0 to b do if ok=1 then for j from 0 to 9 do
%p A248051 if not (j=0 and k=b) then if type(sqrt(trunc(a/10^k)*10^(k+1)+j*10^k+(a mod 10^k)),integer)
%p A248051 then print(n); ok:=0; break; fi; fi; od; fi;
%p A248051 od; od; end: P(10^6);
%t A248051 f[n_] := ! MissingQ@SelectFirst[Rest@Flatten[Outer[Insert[IntegerDigits[n^3], #2, #1] &, Range[IntegerLength[n^3] + 1], Range[0, 9]], 1], IntegerQ@Sqrt@FromDigits@# &];
%t A248051 Select[Range[100], f] (* _Davin Park_, Dec 28 2016 *)
%Y A248051 Cf. A248127, A249853.
%K A248051 nonn,base
%O A248051 1,2
%A A248051 _Paolo P. Lava_, Nov 10 2014
%E A248051 Corrected and extended by _Davin Park_, Dec 26 2016
%E A248051 Extended by _Robert G. Wilson v_, Dec 27 2016