A248051 Numbers whose cubes become squares if some digit is prepended, inserted or appended.
1, 2, 5, 6, 10, 25, 30, 40, 41, 60, 84, 90, 96, 100, 121, 129, 160, 169, 200, 201, 250, 266, 360, 400, 490, 500, 600, 640, 724, 810, 1000, 1025, 1210, 1440, 1690, 1960, 2250, 2500, 2560, 2890, 3000, 3240, 3604, 3610, 4000, 4100, 4410, 4840, 5216, 5290, 5760
Offset: 1
Examples
If n = 1 then n^3 = 1 and if we append a 6 we have sqrt(16) = 4. If n = 2 then n^3 = 8 and if we append a 1 we have sqrt(81) = 9. If n = 5 then n^3 = 125 and if we insert a 2 we get sqrt(1225) = 35. Again, if n = 25 then n^3 = 15625 and we have sqrt(105625) = 325 or sqrt(156025) = 395.
Links
- Robert G. Wilson v, Table of n, a(n) for n = 1..373 (terms 1..100 from Paolo P. Lava, terms 101..144 from Davin Park)
Programs
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Maple
with(numtheory): P:=proc(q) local a,b,j,k,n,ok; for n from 1 to q do a:=n^3; b:=ilog10(a)+1; ok:=1; for k from 0 to b do if ok=1 then for j from 0 to 9 do 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) then print(n); ok:=0; break; fi; fi; od; fi; od; od; end: P(10^6);
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Mathematica
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@# &]; Select[Range[100], f] (* Davin Park, Dec 28 2016 *)
Extensions
Corrected and extended by Davin Park, Dec 26 2016
Extended by Robert G. Wilson v, Dec 27 2016
Comments