A171746 Let f(n) = n + floor(sqrt(n)). Then a(n) is the smallest number of iterations of f on n such that a perfect square is obtained.

3, 2, 1, 5, 2, 4, 1, 3, 7, 2, 4, 6, 1, 3, 5, 9, 2, 4, 6, 8, 1, 3, 5, 7, 11, 2, 4, 6, 8, 10, 1, 3, 5, 7, 9, 13, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 15, 2, 4, 6, 8, 10, 12, 14, 1, 3, 5, 7, 9, 11, 13, 17, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 19, 2, 4, 6, 8, 10, 12, 14, 16, 18, 1, 3, 5

Offset: 1

Author

Neven Juric (neven.juric(AT)apis-it.hr), Oct 07 2010

Keywords

• nonn

Comments

Iterate A028392, starting with n: a(n) is the number of steps until a square will be reached. - Reinhard Zumkeller, Feb 23 2012

Examples

			f(9)=12, f(12)=15, f(15)=18, f(18)=22, f(22)=26, f(26)=31, f(31)=36. The first square number in this sequence 12,15,18,22,26,31,36 is on the seventh place and therefore a(9)=7.
		

References

• Matematicko-fizicki list 1/144, problem 2-2, page 29, (1985-1986).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A010052, A028392.

Programs

Haskell

a171746 = (+ 1) . length . takeWhile (== 0) .
                           map a010052 . tail . iterate a028392
-- Reinhard Zumkeller, Feb 23 2012, Oct 14 2010

Mathematica

f[n_] := Length@ NestWhileList[ # + Floor@Sqrt@# &, n, ! IntegerQ@Sqrt@# || # == n &] - 1; Array[f, 93] (* Robert G. Wilson v, Oct 08 2010 *)

PARI

f(n) = n + sqrtint(n); \\ A028392
a(n) = my(k=1); while (!issquare(n=f(n)), k++); k; \\ Michel Marcus, Nov 06 2022

Formula

From Robert G. Wilson v, Oct 08 2010: (Start)

a(k)=1 for A002061(n): n^2 - n + 1 for n>1;

a(k)=2 for A002522(n): n^2 + 1 for n>1;

a(k)=3 for A014206(n): n^2 + n + 2 for n>1;

a(k)=4 for A059100(n): n^2 + 2 for n>1;

a(k)=5 for A027688(n): n^2 + n + 3 for n>2;

a(k)=6 for A117950(n): n^2 + 3 for n>2;

a(k)=7 for A027689(n): n^2 + n + 4 for n>4;

a(k)=8 for A087475(n): n^2 + 4 for n>3;

a(k)=9 for A027690(n): n^2 + n + 5 for n>4; ... (End)

a(n^2) = 2*n + 1: a(A000290(n)) = A005408(n). - Reinhard Zumkeller, Oct 14 2010