A003059 -id:A003059 - cp's OEIS frontend

A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

1, 10, 46, 130, 265, 463, 799, 1339, 2014, 2780, 3860, 5444, 7301, 9263, 11783, 15263, 19250, 23237, 27893, 34193, 41519, 48701, 56765, 67421, 79484, 91067, 103739, 119855, 138035, 155819, 174923, 198863, 225890, 251444, 277976, 311492, 349122, 384420, 421284

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045851.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045851, A055408, A055415, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341405.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 9)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..38);  # Alois P. Heinz, Feb 10 2021

nmax = 38; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^9/(512 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^9 / (512 * (1 - x)).

a(n^2) = A055408(n).

A341405 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

Original entry on oeis.org

1, 11, 56, 176, 396, 738, 1308, 2268, 3618, 5258, 7449, 10689, 14889, 19609, 25369, 33289, 43154, 53774, 65739, 81339, 100671, 121221, 143421, 171501, 205701, 241283, 278678, 324398, 378998, 435968, 495428, 566468, 650798, 737888, 826083, 930123, 1053323

Offset: 0

Ilya Gutkovskiy, Feb 10 2021

nonn

Partial sums of A045852.

Index entries for sequences related to sums of squares

Cf. A000122, A000606, A003059, A045852, A055409, A055416, A224212, A224213, A302862, A341400, A341401, A341402, A341403, A341404.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(n<0 or k<1, 0,
      b(n, k-1)+add(b(n-j^2, k-1), j=1..isqrt(n))))
    end:
a:= proc(n) option remember; b(n, 10)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 10 2021

nmax = 36; CoefficientList[Series[(1 + EllipticTheta[3, 0, x])^10/(1024 (1 - x)), {x, 0, nmax}], x]

G.f.: (1 + theta_3(x))^10 / (1024 * (1 - x)).

a(n^2) = A055409(n).

A134918 Ceiling(n^(5/3)).

Original entry on oeis.org

1, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 63, 72, 82, 92, 102, 113, 124, 136, 148, 160, 173, 187, 200, 214, 229, 243, 259, 274, 290, 306, 323, 340, 357, 375, 393, 411, 430, 449, 468, 488, 508, 528, 549, 570, 591, 613, 634, 657

Offset: 1

Mohammad K. Azarian, Nov 17 2007

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

Cf. A048766, A038605, A003059, A134914, A129011, A100196, A121536, A134917.

[Ceiling(n^(5/3)): n in [1..50]]; // Vincenzo Librandi, Feb 18 2013

Table[Ceiling[n^(5/3)], {n, 50}] (* Vincenzo Librandi, Feb 18 2013 *)

A135034 Positive integers n repeated 2n-1 times, with a leading a(0) = 0. Also: ceiling of square root of n.

Original entry on oeis.org

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

Offset: 0

William A. Tedeschi, Feb 10 2008

			a(1) = ceiling(sqrt(1)) = 1
a(6) = ceiling(sqrt(6)) = 3

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A005408, A003059 (restriction to positive indices), A000194 (round(sqrt(n))), A000196 (floor(sqrt(n))).

Partial sums of A010052.

Table[Ceiling[Sqrt[n]],{n,0,100}] (* Mohammad K. Azarian, Jun 15 2016 *)
Table[PadRight[{},2n-1,n],{n,0,10}]//Flatten (* Harvey P. Dale, May 15 2022 *)

A135034(n)=ceil(sqrt(n)) \\ M. F. Hasler, Nov 12 2017

from math import isqrt
def A135034(n): return isqrt(n-1)+1 if n else 0 # Chai Wah Wu, Nov 04 2024

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

a(n) = A003059(n), for n >= 1. - R. J. Mathar, Jun 18 2008

Edited and corrected by M. F. Hasler, Nov 12 2017

A174804 a(n) = nceiling(sqrt(n))floor(sqrt(n)).

Original entry on oeis.org

0, 1, 4, 6, 16, 30, 36, 42, 48, 81, 120, 132, 144, 156, 168, 180, 256, 340, 360, 380, 400, 420, 440, 460, 480, 625, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1296, 1554, 1596, 1638, 1680, 1722, 1764, 1806, 1848, 1890, 1932, 1974, 2016, 2401, 2800

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 29 2010

nonn

As a(n^2) = n^4, A000583 is a subsequence. - Bernard Schott, Feb 01 2023

Cf. A000196, A000583, A003059, A038760, A174803.

f[n_]:=n*Floor[Sqrt[n]]*Ceiling[Sqrt[n]];Table[f[n],{n,0,5!}]

a(n) = n*sqrtint(n)*ceil(sqrt(n)); \\ Michel Marcus, Feb 14 2018

from math import isqrt
def A174804(n): return n*(n if (k:=(m:=isqrt(n))**2)==n else k+m) # Chai Wah Wu, Jul 29 2022

a(n) = n*A000196(n)*A003059(n). - Michel Marcus, Feb 14 2018

A295007 Numbers n such that the largest digit of n^2 is 7.

Original entry on oeis.org

24, 26, 42, 52, 61, 69, 74, 76, 82, 84, 85, 88, 124, 131, 132, 144, 154, 165, 166, 174, 181, 189, 194, 218, 224, 226, 234, 239, 240, 260, 265, 266, 269, 271, 274, 275, 276, 319, 326, 356, 371, 376, 384, 415, 416, 418, 419, 420, 421, 448, 455, 466, 474, 476, 520, 521, 524, 525, 526, 552

Offset: 1

M. F. Hasler, Nov 12 2017

			24 is in this sequence because 24^2 = 576 has 7 as largest digit.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A295017 (the corresponding squares), A277959 .. A277961 (same for digit 2 .. 4), A295005 .. A295009 (same for digit 5 .. 9).

Cf. A000290 (the squares).

filter:= proc(n) max(convert(n^2,base,10))=7 end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 19 2019

select( is_A295007(n)=n&&vecmax(digits(n^2))==7 , [0..999]) \\ The "n&&" avoids an error message for n=0.

a(n) = sqrt(A295017(n)), where sqrt = A000196 or A000194 or A003059.

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

A327672 a(n) = Sum_{k=0..n} ceiling(sqrt(k)).

Original entry on oeis.org

0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 260, 268, 276, 284, 292, 300, 308, 316

Offset: 0

Peter Kagey, Sep 21 2019

nonn

Partial sums of A003059.

Given a digraph whose vertices are numbered from 0 to n and in which an edge (u,v) exists iff u < v, a(n) is the maximum number of arcs that can be chosen so that for each vertex j other than 0 and n, the number of chosen arcs whose tail is vertex j equals the number of chosen arcs whose head is vertex j. - Xutong Ding, Dec 12 2023

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A003059, A022554.

Accumulate[Ceiling[Sqrt[Range[0, 60]]]]
Table[(1 + Floor[Sqrt[n]])*(6*n - Floor[Sqrt[n]] - 2*Floor[Sqrt[n]]^2)/6, {n, 0, 100}] (* Vaclav Kotesovec, Dec 26 2023 *)

a(n) = (1 + floor(sqrt(n)))*(6*n - floor(sqrt(n)) - 2*floor(sqrt(n))^2)/6. - Vaclav Kotesovec, Dec 26 2023

A336302 a(n) = n^2 mod ceiling(sqrt(n)).

Original entry on oeis.org

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

Offset: 1

William Phoenix Marcum, Oct 05 2020

A pattern emerges for certain values where ceiling(sqrt(n)) is the same.

Michel Marcus, Table of n, a(n) for n = 1..10000
William Marcum, Desmos graph

Cf. A000290, A003059.

Table[Mod[n^2, Ceiling @ Sqrt[n]], {n, 100}] (* Amiram Eldar, Oct 05 2020 *)

a(n) = lift(Mod(n, ceil(sqrt(n)))^2); \\ Michel Marcus, Oct 05 2020

a(n) = A000290(n) mod A003059(n). - Michel Marcus, Oct 05 2020

a(n^2) = 0. - Michel Marcus, Oct 07 2020

A373461 a(n) = s - t where s = ceiling(sqrt(n*i)), t = sqrt(m), and m = s^2 mod n, for the smallest positive integer i for which m is square.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 5, 2, 7, 8, 3, 4, 9, 6, 11, 4, 3, 14, 15, 4, 5, 16, 3, 6, 19, 6, 21, 4, 9, 24, 5, 6, 25, 26, 9, 4, 29, 6, 31, 12, 5, 34, 35, 6, 7, 10, 9, 14, 39, 12, 5, 8, 9, 44, 45, 6, 47, 48, 7, 8, 5, 12, 51, 20

Offset: 1

Darío Clavijo, Jun 06 2024

nonn

This is "s - t" in Hart's factoring algorithm.

The quantities found have s^2 - t^2 = (s-t)*(s+t) = n*i when n >= 3 and Hart notes that g = gcd(s-t, n) is a nontrivial factor of n (when n is composite).

			For n=9, i=1, s=ceiling(sqrt(9*1))=3 and m=0 then s-floor(sqrt(m))=3-0=3, so a(9)=3.
Also gcd(9, 3) gives a divisor of 3.

S. S. Wagstaff, Jr., The Joy of Factoring, AMS, 2013, pages 119-120.

William B. Hart, A One Line Factoring Algorithm, J. Aust. Math. Soc. 92 (2012), 61-69.

Cf. A003059, A362502.

a(n) = my(i=1,s,t); while(!issquare((s=sqrtint((n*i)-1)+1)^2 % n, &t), i++); s-t;

from sympy.ntheory.primetest import is_square
from sympy.core.power import isqrt
A003059 = lambda n: isqrt((n)-1)+1
def a(n):
  i = 1
  while True:
    s = A003059(n*i)
    if is_square(m:=pow(s,2,n)):
      return s-isqrt(m)
    i+=1
print([a(n) for n in range(1, 69)])

cp's OEIS Frontend

A341404 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_9)^2 <= n.

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A341405 Number of nonnegative solutions to (x_1)^2 + (x_2)^2 + ... + (x_10)^2 <= n.

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A134918 Ceiling(n^(5/3)).

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A135034 Positive integers n repeated 2n-1 times, with a leading a(0) = 0. Also: ceiling of square root of n.

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A174804 a(n) = n*ceiling(sqrt(n))*floor(sqrt(n)).

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A295007 Numbers n such that the largest digit of n^2 is 7.

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

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A174804 a(n) = nceiling(sqrt(n))floor(sqrt(n)).