A173517 -id:A173517 - cp's OEIS frontend

A000037 Numbers that are not squares (or, the nonsquares).

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

N. J. A. Sloane, Simon Plouffe

Note the remarkable formula for the n-th term (see the FORMULA section)!
These are the natural numbers with an even number of divisors. The number of divisors is odd for the complementary sequence, the squares (sequence A000290) and the numbers for which the number of divisors is divisible by 3 is sequence A059269. - Ola Veshta (olaveshta(AT)my-deja.com), Apr 04 2001
a(n) is the largest integer m not equal to n such that n = (floor(n^2/m) + m)/2. - Alexander R. Povolotsky, Feb 10 2008
Union of A007969 and A007970; A007968(a(n)) > 0. - Reinhard Zumkeller, Jun 18 2011
Terms of even numbered rows in the triangle A199332. - Reinhard Zumkeller, Nov 23 2011
If a(n) and a(n+1) are of the same parity then (a(n)+a(n+1))/2 is a square. - Zak Seidov, Aug 13 2012
Theaetetus of Athens proved the irrationality of the square roots of these numbers in the 4th century BC. - Charles R Greathouse IV, Apr 18 2013
4*a(n) are the even members of A079896, the discriminants of indefinite binary quadratic forms. - Wolfdieter Lang, Jun 14 2013

			For example note that the squares 0, 1, 4, 9, 16 are not included.

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 58-60.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 9900 terms from N. J. A. Sloane)
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
A. J. dos Reis and D. M. Silberger, Generating nonpowers by formula, Math. Mag., 63 (1990), 53-55.
Bakir Farhi, An explicit formula generating the non-Fibonacci numbers, arXiv:1105.1127 [math.NT], May 05 2011.
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
S. Kaji, T. Maeno, K. Nuida, and Y. Numata, Polynomial Expressions of Carries in p-ary Arithmetics, arXiv preprint arXiv:1506.02742 [math.CO], 2015-2016.
J. Lambek and L. Moser, Inverse and complementary sequences of natural numbers, Amer. Math. Monthly, 61 (1954), 454-458. doi 10.2307/2308078, see example 4 (includes the formula). [Nicolas Normand (Nicolas.Normand(AT)polytech.univ-nantes.fr), Nov 24 2009]
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Cristinel Mortici, Remarks on Complementary Sequences, Fibonacci Quart. 48 (2010), no. 4, 343-347.
R. D. Nelson, Sequences which omit powers, The Mathematical Gazette, Number 461, 1988, pages 208-211.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Rosetta Code, Sequence of non-squares
J. Scholes, 27th Putnam 1966 Prob. A4
Aaron Snook, Augmented Integer Linear Recurrences, 2012. - From _N. J. A. Sloane_, Dec 19 2012
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Continued Fraction

Cf. A007412, A000005, A000290, A059269, A134986, A087153, A172151, A000196, A049068 (subsequence).
Cf. A242401 (subsequence).
Cf. A086849 (partial sums), A048395.

a000037 n = n + a000196 (n + a000196 n)
-- Reinhard Zumkeller, Nov 23 2011

[n : n in [1..1000] | not IsSquare(n) ];

at:=0; for n in [1..10000] do if not IsSquare(n) then at:=at+1; print at, n; end if; end for;

Maple
```
A000037 := n->n+floor(1/2+sqrt(n));
```

a[n_] := (n + Floor[Sqrt[n + Floor[Sqrt[n]]]]); Table[a[n], {n, 71}] (* Robert G. Wilson v, Sep 24 2004 *)
With[{upto=100},Complement[Range[upto],Range[Floor[Sqrt[upto]]]^2]] (* Harvey P. Dale, Dec 02 2011 *)
a[ n_] :=  If[ n < 0, 0, n + Round @ Sqrt @ n]; (* Michael Somos, May 28 2014 *)

A000037(n):=n + floor(1/2 + sqrt(n))$ makelist(A000037(n),n,1,50); /* Martin Ettl, Nov 15 2012 */

{a(n) = if( n<0, 0, n + (1 + sqrtint(4*n)) \ 2)};

from math import isqrt
def A000037(n): return n+isqrt(n+isqrt(n)) # Chai Wah Wu, Mar 31 2022

from math import isqrt
def A000037(n): return n+(k:=isqrt(n))+int(n>=k*(k+1)+1) # Chai Wah Wu, Jun 17 2024

a(n) = n + floor(1/2 + sqrt(n)).
a(n) = n + floor(sqrt( n + floor(sqrt n))).
A010052(a(n)) = 0. - Reinhard Zumkeller, Jan 26 2010
A173517(a(n)) = n; a(n)^2 = A030140(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000194(n) + n. - Jaroslav Krizek, Jun 14 2009
a(A002061(n)) = a(n^2-n+1) = A002522(n) = n^2 + 1. - Jaroslav Krizek, Jun 21 2009

Edited by Charles R Greathouse IV, Oct 30 2009

A028391 a(n) = n - floor(sqrt(n)).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65

Offset: 0

John Mellor (u15630(AT)snet.net)

Number of nonsquares <= n.
Number of k <= n with an even number of divisors. - Benoit Cloitre, Sep 07 2002
Construct the pyramid
............a(0)
.......a(1).a(2).a(3)
..a(4).a(5).a(6).a(7).a(8).. etc.
Now circle all the primes and the result will be a pattern very similar to the famous Ulam spiral. - Sam Alexander, Nov 14 2003
The sequence floor(n-n^(1/2)) gives the same numbers with a different offset. - Mohammad K. Azarian, R. J. Mathar and M. F. Hasler, Apr 30 2008
The number of nonzero values of floor (j^2/n) taken over 1 <= j <= n-1.
a(n) = A173517(n) iff n is not a square. - Reinhard Zumkeller, Feb 20 2010
a(n) - a(n-1) = 0 if n is a square, otherwise 1. - Robert Israel, Dec 30 2014

B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, pp. 13-40 of Contemporary Design Theory, ed. J. H. Dinizt and D. R. Stinson, Wiley, 1992 (see Theorem 2.7).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Dick Boland, Introduction to the Square Spine Spiral, 2000-2003 [broken link].

Cf. A056847, A000196, A135662-A135665, A166373.

a028391 n = n - a000196 n  -- Reinhard Zumkeller, Oct 28 2012

[n-Floor(Sqrt(n)): n in [0..100]]; // Vincenzo Librandi Dec 31 2014

seq(n - floor(sqrt(n)), n = 0 .. 100); # Robert Israel, Dec 30 2014

f[n_]:=n-Floor[Sqrt[n]];Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)

a(n)=n-sqrtint(n) \\ Charles R Greathouse IV, Jun 28 2013

from math import isqrt
def A028391(n): return n-isqrt(n) # Chai Wah Wu, Jul 28 2022

a(n) = ceiling(n - sqrt(n)), as follows from ceiling(-x) = -floor(x). [Corrected by M. F. Hasler, Feb 21 2010]
a(n) = 2*n - A028392(n). - Reinhard Zumkeller, Oct 28 2012
G.f.: (1+x)/(2*(1-x)^2) - Theta3(0,x)/(2*(1-x)) where Theta3 is a Jacobi theta function. - Robert Israel, Dec 30 2014

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, May 01 2008
Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009
Formula corrected by M. F. Hasler, Feb 21 2010
More terms from Vladimir Joseph Stephan Orlovsky, Mar 29 2010

A320471 a(n) = floor(sqrt(n)) mod ceiling(sqrt(n)).

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Oct 13 2018

nonn

Sequence consists of zeros interleaved with the positive integers, each positive integer k appearing 2k times.

Cf. A000196, A003059, A037213, A049240, A173517.

[Binomial(Ceiling(Sqrt(n)), Floor(Sqrt(n))) - 1: n in [1..100]]; // Vincenzo Librandi, Dec 02 2018

a:= proc(n) modp(floor(sqrt(n)),ceil(sqrt(n))) end: seq(a(n),n=1..100); # Muniru A Asiru, Oct 17 2018

Array[Mod[Floor@ #, Ceiling@ #] &@ Sqrt@ # &, 99] (* or *)
Array[IntegerPart@ # - If[IntegerQ@ #, #, 0] &@ Sqrt@ # &, 99] (* or *)
Flatten@ Array[{0}~Join~ConstantArray[#, 2 #] &, 9] (* Michael De Vlieger, Oct 15 2018 *)

a(n) = sqrtint(n) % (1+sqrtint(n-1)); \\ Michel Marcus, Nov 04 2018

a(n) = sqrtint(n-1) * !issquare(n) \\ David A. Corneth, Nov 04 2018

from math import isqrt
def A320471(n): return 0 if (m:=isqrt(n))**2==n else m # Chai Wah Wu, Jul 29 2022

a(n) = A000196(n) - A037213(n).
a(n) = A000196(n)*A049240(n).
a(n) = A000196(n) mod A003059(n).
a(n) = (n - A173517(n)) - A037213(n)^2.
a(n) = binomial(ceiling(sqrt(n)),floor(sqrt(n))) - 1.
From David A. Corneth, Nov 04 2018: (Start)
a(k^2) = 0.
a(m) = floor(sqrt(m)) for nonsquare m. (End)

Corrected by Michel Marcus, Jun 14 2022

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A000037 Numbers that are not squares (or, the nonsquares).

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A028391 a(n) = n - floor(sqrt(n)).

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A320471 a(n) = floor(sqrt(n)) mod ceiling(sqrt(n)).

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