A000196 Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.

0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Also the integer part of the geometric mean of the divisors of n. - Amarnath Murthy, Dec 19 2001

Number of numbers k (<= n) with an odd number of divisors. - Benoit Cloitre, Sep 07 2002

Also, for n > 0, the number of digits when writing n in base where place values are squares, cf. A007961; A190321(n) <= a(n). - Reinhard Zumkeller, May 08 2011

The least monotonic left inverse of squares, A000290. That is, the lexicographically least nondecreasing sequence a(n) such that a(A000290(n)) = n. - Antti Karttunen, Oct 06 2017

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 73, problem 23.
• Lionel Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.
• Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, p. 28.
• N. J. A. Sloane and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.

Links

• Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
• Krassimir Atanassov, On Some of Smarandache's Problems
• Krassimir Atanassov, On the 100-th, the 101-st and 102-nd Smarandache's problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 94-96.
• Henry Bottomley, Illustration of A000196, A048760, A053186.
• Matthew Hyatt and Marina Skyers, On the Increases of the Sequence floor(k*sqrt(n)), Electronic Journal of Combinatorial Number Theory, Volume 15 (2015), #A17.
• Lionel Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
• Paul Pollack and Joseph Vandehey, Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25)...., The American Mathematical Monthly 122.8 (2015): 757-765.
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
• Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Crossrefs

Cf. A000290, A002162, A003056, A028391, A048760, A048766, A074704, A079051.

Column k=1 of A281871.

Programs

Haskell

import Data.Bits (shiftL, shiftR)
a000196 :: Integer -> Integer
a000196 0 = 0
a000196 n = newton n (findx0 n 1) where
   -- find x0 == 2^(a+1), such that 4^a <= n < 4^(a+1).
   findx0 0 b = b
   findx0 a b = findx0 (a `shiftR` 2) (b `shiftL` 1)
   newton n x = if x' < x then newton n x' else x
                where x' = (x + n `div` x) `div` 2
a000196_list = concat $ zipWith replicate [1,3..] [0..]
-- Reinhard Zumkeller, Apr 12 2012, Oct 23 2010

Julia

a(n) = isqrt(n) # Paul Muljadi, Jun 03 2024

Magma

[Isqrt(n) : n in [0..100]];

Maple

Digits := 100; A000196 := n->floor(evalf(sqrt(n)));

Mathematica

Table[n, {n, 0, 20}, {2n + 1}] //Flatten (* Zak Seidov Mar 19 2011 *)
IntegerPart[Sqrt[Range[0, 110]]] (* Harvey P. Dale, May 23 2012 *)
Floor[Sqrt[Range[0, 99]]] (* Alonso del Arte, Dec 31 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]  - 1) / (2 (1 - x)), {x, 0, n}]; (* Michael Somos, May 28 2014 *)

PARI

{a(n) = if( n<0, 0, sqrtint(n))};

Python

# from http://code.activestate.com/recipes/577821-integer-square-root-function/
def A000196(n):
  if n < 0:
    raise ValueError('only defined for nonnegative n')
  if n == 0:
    return 0
  a, b = divmod(n.bit_length(), 2)
  j = 2**(a+b)
  while True:
    k = (j + n//j)//2
    if k >= j:
      return j
    j = k
print([A000196(n)for n in range(102)])
# Jason Kimberley, Nov 09 2016

Python

from math import isqrt
def a(n): return isqrt(n)
print([a(n) for n in range(102)]) # Michael S. Branicky, Feb 15 2023

Scheme

;; The following implementation uses higher order function LEFTINV-LEASTMONO-NC2NC from my IntSeq-library. It returns the least monotonic left inverse of any strictly growing function (see the comment-section for the definition) and although it does not converge as fast to the result as many specialized integer square root algorithms, at least it does not involve any floating point arithmetic. Thus with correctly implemented bignums it will produce correct results even with very large arguments, in contrast to just taking the floor of (sqrt n).
;; Source of LEFTINV-LEASTMONO-NC2NC can be found under https://github.com/karttu/IntSeq/blob/master/src/Transforms/transforms-core.ss and the definition of A000290 is given under that entry.
(define A000196 (LEFTINV-LEASTMONO-NC2NC 0 0 A000290)) ;; Antti Karttunen, Oct 06 2017

Formula

a(n) = Card(k, 0 < k <= n such that k is relatively prime to core(k)) where core(x) is the squarefree part of x. - Benoit Cloitre, May 02 2002

a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller, Apr 12 2004

From Hieronymus Fischer, May 26 2007: (Start)

a(n) = Sum_{k=1..n} A010052(k).

G.f.: g(x) = (1/(1-x))*Sum_{j>=1} x^(j^2) = (theta_3(0, x) - 1)/(2*(1-x)) where theta_3 is a Jacobi theta function. (End)

a(n) = floor(A000267(n)/2). - Reinhard Zumkeller, Jun 27 2011

a(n) = floor(sqrt(n)). - Arkadiusz Wesolowski, Jan 09 2013

Sum_{n>0} 1/a(n)^s = 2*zeta(s-1) + zeta(s), where zeta is the Riemann zeta function. - Enrique Pérez Herrero, Oct 15 2013

From Wesley Ivan Hurt, Dec 31 2013: (Start)

a(n) = Sum_{i=1..n} (A000005(i) mod 2), n > 0.

a(n) = (1/2)*Sum_{i=1..n} (1 - (-1)^A000005(i)), n > 0. (End)

a(n) = sqrt(A048760(n)), n >= 0. - Wolfdieter Lang, Mar 24 2015

a(n) = Sum_{k=1..n} floor(n/k)*lambda(k) = Sum_{m=1..n} Sum_{d|m} lambda(d) where lambda(j) is Liouville lambda function, A008836. - Geoffrey Critzer, Apr 01 2015

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, May 02 2023