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 A000196 Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times. 331
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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 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. 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 EXAMPLE 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 + ... 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 *) PROG (Magma) [Isqrt(n) : n in [0..100]]; (PARI) {a(n) = if( n<0, 0, floor(sqrt(n)))}; (PARI) {a(n) = if( n<0, 0, sqrtint(n))}; (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 (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 CROSSREFS Cf. A000290, A002162, A003056, A028391, A048760, A048766, A074704, A079051. Column k=1 of A281871. Sequence in context: A350088 A178786 A350174 * A111850 A343265 A059396 Adjacent sequences: A000193 A000194 A000195 * A000197 A000198 A000199 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane STATUS approved

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Last modified April 20 06:23 EDT 2024. Contains 371799 sequences. (Running on oeis4.)