A105209 -id:A105209 - cp's OEIS frontend

A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 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, 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, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

Define the oblong root obrt(x) to be the (larger) solution of y * (y+1) = x; i.e., obrt(x) = sqrt(x+1/4) - 1/2. So obrt(x) is an integer iff x is an oblong number (A002378). Then a(n) = ceiling(obrt(n)). - Franklin T. Adams-Watters, Jun 24 2015
From Wolfdieter Lang, Mar 12 2019: (Start)
The general Pell equation is related to the non-reduced form F(n) = Xvec^T A(n) Xvec = x^2 - D(n)*y^2 with D(n) = A000037(n) (D not a square), Xvec = (x,y)^T (T for transposed) and A(n) = matrix[[1,0], [0,-D(n)]]. The discriminant of F(n) = [1, 0, -D(n)] is 4*D(n).
The first reduced form appears after two applications of an equivalence transformation A' = R^T A R obtained with R = R(t) = matrix([0, -1], [1, t]), namely first with t = 0, leading to the still not reduced form [-D, 0, 1], and then with t = ceiling(f(4*D(n))/2 - 1), where f(4*D(n)) = ceiling(2*sqrt(D(n))). This can be shown to be a(n), which is also D(n) - n, for n >= 1 (see a formula below).
This leads to the reduced form FR(n) = [1, 2*a(n), -(D(n) - a(n)^2)] = [1, 2*a(n), -(n - a(n)*(a(n) - 1))]. Example: n = 5, a(5) = 2: D(5) = 7 and FR(5) = [1, 4, -3].  (End)

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 3*x^10 + ...

Titu Andreescu, Dorin Andrica, and Zuming Feng, 104 Number Theory Problems, Birkhäuser, 2006, 59-60.
B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 78, Entry 24.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.
G. Gutin, Problem 913 (BCC20.5), Mediated digraphs, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 2002), 559-564.
Michael Somos, Sequences used for indexing triangular or square arrays.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000037, A002024, A002061, A002378, A053187, A105209, A259351.
Partial sums of A005369.
Cf. A000196 (floor(sqrt(n))), A003059 (ceiling(sqrt(n))).

a000194 n = a000194_list !! (n-1)
a000194_list = concat $ zipWith ($) (map replicate [2,4..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

Digits := 100; f := n->round(evalf(sqrt(n))); [ seq(f(n), n=0..100) ];
# More efficient:
a := n -> isqrt(n): seq(a(n), n=0..98); # Peter Luschny, Mar 13 2019

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2]; (* Enrique Pérez Herrero, Apr 14 2010 *)
Flatten[Table[PadRight[{}, 2 n, n], {n, 10}]] (* Harvey P. Dale, Nov 16 2011 *)
CoefficientList[Series[x QPochhammer[-x^2, x^4] QPochhammer[x^8, x^8]/(1 - x), {x, 0, 50}], x] (* Eric W. Weisstein, Jan 10 2024 *)

{a(n) = ceil( sqrtint(4*n) / 2)}; /* Michael Somos, Feb 11 2004 */

a(n)=(sqrtint(4*n) + 1)\2 \\ Charles R Greathouse IV, Jun 08 2020

apply( {A000194(n)=sqrtint(4*n)\/2}, [0..99]) \\ M. F. Hasler, Jun 22 2024

from math import isqrt
def A000194(n): return (m:=isqrt(n))+int(n-m*(m+1)>=1) # Chai Wah Wu, Jul 30 2022

a(n) = A000037(n) - n.
G.f.: x * f(x^2, x^6)/(1-x) where f(,) is Ramanujan's two-variable theta function. - Michael Somos, May 31 2000
a(n) = a(n - 2*a(n - a(n-1))) + 1. - Benoit Cloitre, Oct 27 2002
a(n+1) = a(n) + A005369(n).
a(n) = floor((1/2)*(1 + sqrt(4*n - 3))). - Zak Seidov, Jan 18 2006
a(n) = A000037(n) - n. - Jaroslav Krizek, Jun 14 2009
a(n) = floor(A027434(n)/2). - Gregory R. Bryant, Apr 17 2013
From Mikael Aaltonen, Jan 17 2015: (Start)
a(n) = floor(sqrt(n) + 1/2).
a(n) = sqrt(A053187(n)). (End)
a(0) = 0, and a(n) = k for k from the closed interval [k^2 - k + 1, k*(k+1)] = [A002061(k), A002378(k)], for k >= 1. See A053187. - Wolfdieter Lang, Mar 12 2019
a(n) = floor(2*sqrt(n)) - floor(sqrt(n)). - Ridouane Oudra, Jun 08 2020
Sum_{n>=1} 1/a(n)^s = 2*zeta(s-1), for s > 2 (Borwein, 1994). - Amiram Eldar, Oct 31 2020

Additional comments from Michael Somos, May 31 2000
Edited by M. F. Hasler, Mar 01 2014
Initial 0 added by N. J. A. Sloane, Nov 13 2017

A147773 a(n) = round((n^n)^(1/3)).

Original entry on oeis.org

1, 2, 3, 6, 15, 36, 94, 256, 729, 2154, 6583, 20736, 67156, 223150, 759375, 2642246, 9387369, 34012224, 125537306, 471556032, 1801088541, 6989288907, 27536796143, 110075314176, 446169698824, 1832746290156, 7625597484987, 32122422687591

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 12 2008

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A105209, A000312.

lst={};Do[AppendTo[lst,Round[(n^n)^(1/3)]],{n,40}];lst

from gmpy2 import iroot_rem
def A147773(n):
    i, j = iroot_rem(n**n,3)
    return int(i+int(8*j >= 6*i*(2*i+1)+1)) # Chai Wah Wu, Aug 16 2016

a(n) = A105209(A000312(n)). - Michel Marcus, Aug 19 2016

A321028 a(n) = 6 + round(n^3) - (minimal number of squares in a dissection of an (n) X (n+1) oblong into squares).

Original entry on oeis.org

5, 4, 3, 3, 3, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Ed Pegg Jr, Oct 26 2018

After a(18) = 2, all terms through a(387) are in (-1,0,1). The first known term outside of this range is a(969) >= 2.

Amiram Eldar, Table of n, a(n) for n = 1..387 (calculated from the b-file at A279317)
Bertram Felgenhauer, Filling Rectangles with Integer-Sided Squares.
Ed Pegg Jr, Minimally Squared Rectangles.
Ed Pegg Jr, Oblongs into minimal squares, StackExchange, Dec 13 2016.

Cf. A105209, A279317.

a(n) = 6 + A105209(n) - A279317(n).

A338478 Let b be an odd function such that b(0) = 0, b(1) = 1, and for any n > 1 such that 3^x < 2*n < 3^(x+1) for some x > 0, b(n) = b(3^x-n) - 3^x; a(n) = abs(b(n)) for any n >= 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 13, 12, 11, 8, 9, 10, 7, 6, 5, 32, 33, 34, 37, 36, 35, 38, 39, 40, 31, 30, 29, 26, 27, 28, 25, 24, 23, 14, 15, 16, 19, 18, 17, 20, 21, 22, 103, 102, 101, 98, 99, 100, 97, 96, 95, 104, 105, 106, 109, 108, 107, 110, 111, 112, 121, 120, 119, 116

Offset: 0

Rémy Sigrist, Oct 29 2020

nonn

This sequence is a self-inverse permutation of the nonnegative integers.

It is possible to build a continuous injective complex-valued function of a real-variable, say f, such that Im(f(r)) = 0 iff r is an integer and for any n in Z, f(n) = b(n) (see illustration in Links section).

			For n = 3:
- we have 3^1 < 2*3 < 3^(1+1),
- so b(3) = b(3 - 3) - 3 = 0 - 3 = -3,
- a(3) = abs(b(3)) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Representation in the complex plane of a function f as described in Comments section (the corresponding curve divides the complex plane into two simply connected regions rendered with different colors)
Index entries for sequences that are permutations of the natural numbers

Cf. A000244, A003462, A024023, A034472, A105209.

b(n) = { if (n<0,  return (-b(-n)), n==0, return (0), n==1, return (1), for (x=1, oo, my (w=3^x, h=w\2); if (w<2*n && 2*n<3*w, return (b(w-n)-w)))) }
a(n) = abs(b(n))

a(n) = n iff abs(n - 3^x) <= 1 for some x >= 0.

A339276 Nearest integer to the fourth root of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Amiram Eldar, Dec 13 2020

			a(1) = 1 since 1^(1/4) = 1.
a(6) = 2 since 6^(1/4) = 1.565... and its nearest integer is 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.

Cf. A000194, A105209, A255270.

Mathematica
```
Table[Round[Surd[n, 4]], {n, 1, 100}]
```

from sympy import integer_nthroot
def A339276(n): return (m:=integer_nthroot(n,4)[0])+((n<<4)>=((m<<1)+1)**4) # Chai Wah Wu, Jun 06 2025

Sum_{n>=1} 1/a(n)^s = 4*zeta(s-3) + zeta(s-1), for s>4 (Borwein, 1994).

cp's OEIS Frontend

A000194 n appears 2n times, for n >= 1; also nearest integer to square root of n.

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A147773 a(n) = round((n^n)^(1/3)).

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A321028 a(n) = 6 + round(n^3) - (minimal number of squares in a dissection of an (n) X (n+1) oblong into squares).

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A338478 Let b be an odd function such that b(0) = 0, b(1) = 1, and for any n > 1 such that 3^x < 2*n < 3^(x+1) for some x > 0, b(n) = b(3^x-n) - 3^x; a(n) = abs(b(n)) for any n >= 0.

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Comments

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PARI

Formula

A339276 Nearest integer to the fourth root of n.

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Author

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Examples

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Programs

Mathematica

Python

Formula