A135665 -id:A135665 - cp's OEIS frontend

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

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
Thomas Bloom, Problem 121, Erdős Problems.
Dick Boland, Introduction to the Square Spine Spiral, 2000-2003.
Terence Tao, Erdős problem database, see no. 121.

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

A135668 a(n) = ceiling(n + sqrt(n)).

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 1

Mohammad K. Azarian, Nov 25 2007

Complement of A002061. - Kieren MacMillan, Dec 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A134914, A135660, A135661, A135662, A135663, A135664, A135665, A028391, A002061.

[Ceiling(n + n^(1/2)): n in [1..100]]; // Vincenzo Librandi, Feb 16 2013

Table[Ceiling[n + n^(1/2)], {n, 100}] (* Vincenzo Librandi, Feb 16 2013 *)

from sympy import integer_nthroot
def A135668(n): return n+(a:=integer_nthroot(n,2))[0]+(not a[1]) # Chai Wah Wu, Aug 26 2024

A135671 a(n) = ceiling(n - n^(2/3)).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 44, 45, 46

Offset: 1

Mohammad K. Azarian, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135669, A135670.

[Ceiling(n - n^(2/3)): n in [1..100]]; // Vincenzo Librandi, Feb 16 2013

Table[Ceiling[n - n^(2/3)], {n, 100}] (* Vincenzo Librandi, Feb 16 2013 *)

a(n)=n-sqrtnint(n^2,3) \\ Charles R Greathouse IV, Sep 02 2015

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A135672 a(n) = floor(n - n^(2/3)).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45

Offset: 1

Mohammad K. Azarian, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135669, A135670, A135671.

[Floor(n-n^(2/3)): n in [1..100]]; // Vincenzo Librandi, Feb 18 2013

Table[Floor[n - n^(2/3)], {n, 100}] (* Vincenzo Librandi, Feb 18 2013 *)

a(n)=n-sqrtnint(n^2-1,3)-1 \\ Charles R Greathouse IV, Sep 02 2015

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A135673 Ceiling(n + n^(2/3)).

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

Mohammad K. Azarian, Nov 25 2007

			a(6) = 10; ceiling(6 + 6^(2/3)) = ceiling(9.30192...) = 10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135669, A135670, A135671, A135672.

[Ceiling(n + n^(2/3)): n in [1..100]]; // Vincenzo Librandi, Feb 16 2013

A135673:=n->ceil(n + n^(2/3)); seq(A135673(k), k=1..100); # Wesley Ivan Hurt, Nov 01 2013

Table[Ceiling[n+n^(2/3)],{n,60}] (* Harvey P. Dale, Nov 17 2012 *)

More terms from Wesley Ivan Hurt, Nov 01 2013

A135674 Floor(n+n^(2/3)).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74

Offset: 1

Mohammad K. Azarian, Nov 25 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135669, A135670, A135671, A135672, A135673.

[Floor(n+n^(2/3)): n in [1..100]]; // Vincenzo Librandi, Feb 18 2013

Table[Floor[n + n^(2/3)], {n, 100}] (* Vincenzo Librandi, Feb 18 2013 *)

A135675 a(n) = ceiling(n^(4/3) - n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 18, 20, 22, 25, 27, 30, 32, 35, 37, 40, 43, 46, 49, 52, 54, 58, 61, 64, 67, 70, 73, 77, 80, 83, 87, 90, 94, 97, 101, 104, 108, 112, 116, 119, 123, 127, 131, 135, 139, 143, 147, 151, 155

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135671, A135672, A135673, A135674.

[Ceiling(n^(4/3) - n): n in [1..100]]; // Vincenzo Librandi, Feb 16 2013

Table[Ceiling[n^(4/3) - n], {n, 100}] (* Vincenzo Librandi, Feb 16 2013 *)

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A135676 a(n) = floor(n^(4/3) - n).

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 24, 26, 29, 31, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 79, 82, 86, 89, 93, 96, 100, 103, 107, 111, 115, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135671, A135672, A135673, A135674, A135675.

[Floor(n^(4/3)-n): n in [1..100]]; // Vincenzo Librandi, Feb 18 2013

Table[Floor[n^(4/3) - n], {n, 100}] (* Vincenzo Librandi, Feb 18 2013 *)

a(n)=sqrtnint(n^4,3)-n \\ Charles R Greathouse IV, Feb 07 2025

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A135677 a(n) = ceiling(n^(4/3)+n).

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 21, 24, 28, 32, 36, 40, 44, 48, 52, 57, 61, 66, 70, 75, 79, 84, 89, 94, 99, 104, 108, 114, 119, 124, 129, 134, 139, 145, 150, 155, 161, 166, 172, 177, 183, 188, 194, 200, 206, 211, 217, 223, 229, 235, 241

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660, A135661, A135662, A135663, A135664, A135665, A028391, A135668, A135671, A135672, A135673, A135674, A135675, A135676.

[Ceiling(n^(4/3)+n): n in [1..100]]; // Vincenzo Librandi, Feb 16 2013

Table[Ceiling[n^(4/3) + n], {n, 100}] (* Vincenzo Librandi, Feb 16 2013 *)

A135678 Floor(n^(4/3)+n).

Original entry on oeis.org

2, 4, 7, 10, 13, 16, 20, 24, 27, 31, 35, 39, 43, 47, 51, 56, 60, 65, 69, 74, 78, 83, 88, 93, 98, 103, 108, 113, 118, 123, 128, 133, 138, 144, 149, 154, 160, 165, 171, 176, 182, 187, 193, 199, 205, 210, 216, 222, 228, 234, 240

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A135660; A135661; A135662; A135663; A135664; A135665; A028391; A135668; A135671; A135672; A135673 ; A135674, A135675; A135676; A135677.

[Floor(n^(4/3)+n): n in [1..100]]; // Vincenzo Librandi, Feb 18 2013

Table[Floor[n^(4/3) + n], {n, 100}] (* Vincenzo Librandi, Feb 18 2013 *)

cp's OEIS Frontend

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

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A135668 a(n) = ceiling(n + sqrt(n)).

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A135671 a(n) = ceiling(n - n^(2/3)).

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A135672 a(n) = floor(n - n^(2/3)).

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A135673 Ceiling(n + n^(2/3)).

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A135674 Floor(n+n^(2/3)).

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Magma

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A135675 a(n) = ceiling(n^(4/3) - n).

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