A025224 -id:A025224 - cp's OEIS frontend

A022554 a(n) = Sum_{k=0..n} floor(sqrt(k)).

0, 1, 2, 3, 5, 7, 9, 11, 13, 16, 19, 22, 25, 28, 31, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

Partial sums of A000196. - Michel Marcus, Mar 01 2016
It seems that a(47) = 197 is the largest prime in this sequence. Tested for n <= 10^11. - Hugo Pfoertner, Oct 26 2020 [This is true. See A214036]
By drawing a picture of the sum Integral_{x=0..n} ceiling(sqrt(x)) dx, one easily sees that it is equal to n*m - Sum_{k=1..m} (k^2 - 1) with m = floor(sqrt(n)), whence the formula. - M. F. Hasler, Apr 23 2022
We can prove that 197 is the largest prime in this sequence. From the formula, 6*a(n) = m*(6n - 2m^2 - 3m + 5) where m = floor(sqrt(n)). Therefore 6*a(n) is divisible by m, which means that a(n) is divisible by A060789(m). For n>48, we have m>6, so A060789(m)>1, so a(n) is not prime; testing all n up to 48, we see that a(47)=197 is the last prime. - Mikhail Lavrov, Dec 09 2023

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 11*x^7 + 13*x^8 + 16*x^9 + ...

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd Edition, Addison-Wesley, 1994, Eq. 3.27 on page 87.
D. E. Knuth, The Art of Computer Programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 43 of section 1.2.4.
K. H. Rosen, Discrete Mathematics and Its Application, 6th Edition, McGraw-Hill, 2007, Ex. 25 of section 2.4.

David A. Corneth, Table of n, a(n) for n = 0..9999 (first 1001 terms from G. C. Greubel)
M. Griffiths, More sums involving the floor function, Math. Gaz., 86 (2002), 285-287.
Michael Penn, Wringing out one more result., YouTube video, 2021.

Cf. A000196 (first differences), A025224, A214036, A000330.

[&+[Floor(Sqrt(k)): k in [0..n]]: n in [0..50]]; // G. C. Greubel, Feb 26 2018

seq(add(floor(sqrt(k)), k=0..n), n=0..59);

Accumulate[Floor[Sqrt[Range[0,60]]]] (* Harvey P. Dale, Feb 16 2011 *)
Table[Sum[Floor[Sqrt[i]], {i,0,n}], {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

a(n)=sum(k=1,n,sqrtint(k)) \\ Charles R Greathouse IV, Jan 12 2012

a(n)=my(k=sqrtint(n));k*(n-(2*k+5)/6*(k-1)) \\ Charles R Greathouse IV, Jan 12 2012

from math import isqrt
def A022554(n): return (m:=isqrt(n))*(m*(-(m<<1)-3)+6*n+5)//6 # Chai Wah Wu, Aug 03 2022

a(0)=0, a(1)=1; a(n) = 2*a(n-1) - a(n-2) if n is not a perfect square; a(n) = 2*a(n-1) - a(n-2) + 1 if n is a perfect square.
a(n) = floor(sqrt(n)) * (n-1/6*(2*floor(sqrt(n))+5)*(floor(sqrt(n))-1)). - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
a(n) = (2/3)*n^(3/2) - (1/2)*n + O(sqrt(n)). - Charles R Greathouse IV, Jan 12 2012
G.f.: Sum_{k>=1} x^(k^2)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016
a(n) = m*(n - m^2/3 - m/2 + 5/6) where m = floor(sqrt(n)). - M. F. Hasler, Apr 23 2022
a(n) = n*t(n) - A000330(t(n)), where t(n) = floor(sqrt(n)). - Ridouane Oudra, Mar 05 2025

More terms from Yong Kong (ykong(AT)curagen.com), Mar 10 2001

A282168 a(n) is the minimal sum of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 29, 33, 37, 41, 45, 49, 53, 57, 62, 67, 72, 77, 82, 87, 92, 97, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 273, 281, 289, 297, 305, 313, 321, 329, 337, 345, 353

Offset: 1

Peter Kagey, Feb 07 2017

nonn

This sequence shares first 12 terms with A025224, but then they diverge: a(13) = 33 > 32 = A025224(13).

We seem to have a(n) = a(n-1) + a(n-2) - a(n-3) + d(n), where d(n) is 0 or 1. Compare to A282166. - Max Alekseyev, Jun 13 2025

			[1,2,3,1,2] is invalid because the substring [1,2] appears twice.
[1,2,1] is invalid because the substring [1,2] appears twice (once forward and once backward).
a(1)  = 1   via [1];
a(2)  = 2   via [1,1];
a(3)  = 4   via [1,1,2];
a(4)  = 6   via [1,1,2,2];
a(5)  = 8   via [1,1,2,3,1];
a(6)  = 10  via [1,1,2,2,3,1];
a(7)  = 13  via [1,1,2,2,3,3,1];
a(8)  = 16  via [1,1,2,2,3,1,4,2];
a(9)  = 19  via [1,1,2,2,3,3,1,4,2];
a(10) = 22  via [1,1,2,2,3,1,4,2,5,1];
a(11) = 25  via [1,1,2,2,3,3,1,4,2,5,1];
a(12) = 29  via [1,1,2,2,3,3,1,4,4,2,5,1].

Cf. A259280, A282166, A282167, A282193.

Edited and terms a(13) onward added by Max Alekseyev, Feb 05 2025

A174058 Round(Sum_{k=1..n} {sqrt(k)}).

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 11, 13, 16, 19, 22, 26, 29, 33, 37, 40, 44, 49, 53, 57, 62, 66, 71, 76, 81, 86, 91, 96, 101, 107, 112, 118, 123, 129, 135, 141, 147, 153, 159, 165, 172, 178, 184, 191, 198, 204, 211, 218, 225, 232, 239, 246, 253, 261, 268, 275, 283, 290, 298, 306

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A016040, A025224

s=0;lst={};Do[s+=Sqrt[n];AppendTo[lst,Round[s]],{n,0,6!}];lst
Accumulate[Sqrt[Range[0,60]]]//Round (* Harvey P. Dale, Oct 16 2018 *)

A137262 Floor of sum of the first 10^n square roots.

Original entry on oeis.org

1, 22, 671, 21097, 666716, 21082008, 666667166, 21081852648, 666666671666, 21081851083600, 666666666716666, 21081851067947309, 666666666667166666, 21081851067790776685, 666666666666671666666, 21081851067789211358047, 666666666666666716666666, 21081851067789195704773173

Offset: 0

Cino Hilliard, Mar 12 2008

nonn

			The first 10^0 square roots is 1. The sum of the first 10^1 square roots is 22.468278186... . So 22 is the second entry in the sequence.

Amiram Eldar, Table of n, a(n) for n = 0..666

Cf. A025224, A195284.

a[n_] := Floor[Sum[Sqrt[j],{j,10^n}]];Array[a,17,0] (* James C. McMahon, May 17 2025 *)
a[n_] := Floor[HarmonicNumber[10^n, -1/2]]; Array[a, 20, 0] (* Amiram Eldar, Jul 18 2025 *)

g2(n,p=2) = for(j=0,n,s=0;for(x=0,10^j,s+=x^(1/p)); print1(floor(s)", "))

From Amiram Eldar, Jul 18 2025: (Start)
a(n) = A025224(10^n).
a(n) ~ (2/3) * 10^(3*n/2). (End)

a(10)-a(16) from James C. McMahon, May 17 2025

a(17) from Amiram Eldar, Jul 18 2025

A174059 a(n) = ceiling(Sum_{k=1..n} sqrt(k)).

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 11, 14, 17, 20, 23, 26, 30, 33, 37, 41, 45, 49, 53, 58, 62, 67, 71, 76, 81, 86, 91, 96, 102, 107, 113, 118, 124, 130, 135, 141, 147, 153, 160, 166, 172, 179, 185, 192, 198, 205, 212, 219, 225, 232, 240, 247, 254, 261, 269, 276, 283, 291, 299, 306

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A025224, A174058

map(ceil,ListTools:-PartialSums([seq((sqrt(k)),k=0..100)])); # Robert Israel, May 06 2019

s=0;lst={};Do[s+=Sqrt[n];AppendTo[lst,Ceiling[s]],{n,0,6!}];lst
Ceiling[Accumulate[Sqrt[Range[0,60]]]] (* Harvey P. Dale, Aug 29 2016 *)

a(n) = 2/3*n^(3/2) + 1/2*n^(1/2) + O(1). It appears that the absolute value of the difference is always less than 1. - Robert Israel, May 06 2019

Offset corrected by Robert Israel, May 06 2019

A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 18, 23, 27, 32, 38, 43, 49, 55, 62, 68, 75, 82, 90, 97, 105, 113, 121, 130, 138, 147, 156, 166, 175, 185, 194, 204, 214, 225, 235, 246, 257, 267, 279, 290, 301, 313, 325, 336, 349, 361, 373, 385, 398

Offset: 1

Balarka Sen, Apr 30 2013

The fact that a(n)/n diverges (it is greater than sqrt(n)) implies sum_{k>=1} 1/sqrt(k) is not Cesaro summable.

Balarka Sen, Table of n, a(n) for n = 1..500

Cf. A022819, A025224.

for(n=1,100,print1(floor(sum(i=1,n,sum(j=1,i,1/sqrt(j))))","))

a(n)=sum(j=1,n,(n+1-j)/sqrt(j))\1 \\ Charles R Greathouse IV, May 02 2013

a(n) ~ 2*Sum_{k=1..n} sqrt(k) ~ (4/3) n^(3/2).

A338277 Greatest integer whose square root is less than or equal to Sum_{j=0..n} sqrt(j).

Original entry on oeis.org

0, 1, 5, 17, 37, 70, 117, 181, 265, 372, 504, 664, 855, 1079, 1339, 1637, 1977, 2361, 2791, 3271, 3802, 4388, 5032, 5735, 6501, 7333, 8232, 9202, 10245, 11364, 12562, 13841, 15204, 16654, 18193, 19824, 21549, 23372, 25295, 27321, 29451, 31690, 34040, 36502, 39081, 41778, 44597, 47539, 50609, 53807

Offset: 0

Robert G. Wilson v, Oct 21 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A025224.

f:= n -> floor(add(sqrt(i),i=1..n)^2):
map(f, [$0..100]); # Robert Israel, Oct 28 2020

a[n_] := Floor[(Sum[ Sqrt[k], {k, 0, n}])^2]; Array[a, 50, 0]

a(n) = floor(sum(j=0, n, sqrt(j))^2); \\ Michel Marcus, Oct 26 2020

a(n) ~ (4/9)*n^3 + (2/3)*n^2 + (4*zeta(-1/2)/3)*n^(3/2) + (11/36)*n + zeta(-1/2)*sqrt(n). - Robert Israel, Oct 28 2020

A352077 a(n) = floor( Sum_{k=1..n} k^(1/3) ).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 12, 14, 16, 19, 21, 23, 26, 28, 31, 33, 36, 39, 41, 44, 47, 50, 53, 56, 58, 61, 65, 68, 71, 74, 77, 80, 83, 87, 90, 93, 97, 100, 104, 107, 110, 114, 117, 121, 125, 128, 132, 136, 139, 143, 147, 150, 154, 158, 162, 166, 170, 173, 177, 181, 185, 189, 193, 197

Offset: 0

Robert G. Wilson v, Mar 02 2022

			a(6) = 8 because 1^(1/3) + 2^(1/3) + 3^(1/3) + 4^(1/3) + 5^(1/3) + 6^(1/3) = 8.81667... .

Snehal Shekatkar, On the sum of the r'th roots of first n natural numbers, arXiv:1204.0877 [math.NT], 2012-2013.

Cf. A031876, A025224.

a[n_] := Floor[ HarmonicNumber[n, -1/3]]; Array[ a, 66, 0]

a(n) = floor(sum(k=0, n, k^(1/3))); \\ Michel Marcus, Mar 02 2022

A025199 a(n) = floor(floor(S2)/floor(S1)), where S2 and S1 are, respectively, the 2nd and first elementary symmetric functions of {sqrt(k), k = 1,2,...,n}.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 15, 17, 18, 20, 23, 25, 27, 29, 31, 34, 36, 38, 41, 43, 46, 48, 51, 54, 57, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 90, 93, 96, 99, 103, 106, 110, 113, 116, 120, 124, 127, 131, 135, 139, 142, 146, 150

Offset: 2

Clark Kimberling

nonn

a(n) = floor(A025193(n) / A025224(n)). - Sean A. Irvine, Aug 16 2019

Title improved by Sean A. Irvine, Aug 16 2019

A025200 a(n) = floor(floor(S3)/floor(S1)), where S3 and S1 are, respectively, the 3rd and first elementary symmetric functions of {sqrt(k), k = 1,2,...,n}.

Original entry on oeis.org

0, 2, 5, 11, 18, 28, 42, 60, 82, 107, 140, 176, 218, 267, 324, 389, 455, 538, 622, 726, 830, 945, 1072, 1211, 1363, 1513, 1692, 1868, 2076, 2282, 2502, 2759, 3013, 3283, 3572, 3854, 4178, 4523, 4860, 5244, 5621, 6048, 6467, 6907, 7370, 7890, 8400, 8896, 9454, 10037, 10647, 11241, 11903, 12594, 13268, 13968

Offset: 2

Clark Kimberling

nonn

Cf. A025194, A025224.

a(n) = floor(A025194(n) / A025224(n)). - Sean A. Irvine, Aug 16 2019

Title improved and more terms from Sean A. Irvine, Aug 16 2019

cp's OEIS Frontend

A022554 a(n) = Sum_{k=0..n} floor(sqrt(k)).

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A282168 a(n) is the minimal sum of a positive integer sequence of length n with no duplicate substrings (forward or backward) of length greater than 1.

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A174058 Round(Sum_{k=1..n} {sqrt(k)}).

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A137262 Floor of sum of the first 10^n square roots.

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A174059 a(n) = ceiling(Sum_{k=1..n} sqrt(k)).

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A225154 Floor(Sum_{i=1..n} (Sum_{j=1..i} sqrt(1/j))).

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A338277 Greatest integer whose square root is less than or equal to Sum_{j=0..n} sqrt(j).

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