A022554 -id:A022554 - cp's OEIS frontend

A025224 a(n) = floor(Sum_{k=1..n} sqrt(k)).

0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 29, 32, 36, 40, 44, 48, 52, 57, 61, 66, 70, 75, 80, 85, 90, 95, 101, 106, 112, 117, 123, 129, 134, 140, 146, 152, 159, 165, 171, 178, 184, 191, 197, 204, 211, 218, 224, 231, 239, 246, 253, 260, 268, 275, 282, 290, 298, 305, 313, 321, 329, 337

Offset: 0

Clark Kimberling

nonn

Shekatkar took Ramanujan's formula for sum of the square roots of first n natural numbers, and generalized to include r-th roots where r is any real number greater than 1, using simple properties of Riemann integrable functions. - Jonathan Vos Post, Apr 05 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. Ramanujan, On the sum of the square roots of the first n natural numbers, Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
S. Ramanujan, On the sum of the square roots of the first n natural numbers; Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
Snehal Shekatkar, On the sum of the r'th roots of first n natural numbers, arXiv:1204.0877 [math.NT], 2012-2013.

Cf. A022554, A352077.

s=0;lst={};Do[s+=Sqrt[n];AppendTo[lst,c=Floor[s]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
Floor@HarmonicNumber[Range[0, 30], -1/2] (* Vladimir Reshetnikov, Nov 07 2015 *)
Floor[Accumulate[Sqrt[Range[0,70]]]] (* Harvey P. Dale, Apr 23 2022 *)

a(n) = floor(sum(k=0, n, sqrt(k))); \\ Michel Marcus, Mar 01 2016

a(n) ~= floor ((4n + 3)sqrt(n)/6 - exp(-Pi / 2)). - Charles R Greathouse IV, Jul 29 2007. Corrected by Carl R. White, Jan 22 2009

A031876 a(n) = Sum_{k=0..n} floor(k^(1/3)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 160

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

K. H. Rosen, Discrete Mathematics and Its Application, 6th Edition, McGraw-Hill, 2007, Ex. 26 of section 2.4.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A022554.

Accumulate[Floor[Surd[Range[0,70],3]]] (* Harvey P. Dale, Nov 03 2013 *)
Table[Sum[Floor[i^(1/3)], {i,0,n}], {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

a(n) = sum(k=1, n, sqrtnint(k, 3)); \\ Michel Marcus, Mar 12 2016

a(n)=my(t=sqrtnint(n,3)); t*(4*n-t^3-2*t^2-t+4)/4 \\ Charles R Greathouse IV, Aug 23 2017

a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) - a(n-2) if n not a perfect cube, else a(n) = 2*a(n-1) - a(n-2) + 1 if n is a perfect cube.
a(n) = -1/4*floor(n^(1/3))*(floor(n^(1/3))^3+2*floor(n^(1/3))^2+floor(n^(1/3))-4*(n+1)). - John M. Campbell, Mar 22 2016
G.f.: Sum_{k>=1} x^(k^3)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016
a(n) = (3/4)*n^(4/3) + O(n). - Charles R Greathouse IV, Aug 23 2017

A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

Original entry on oeis.org

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269, 272

Offset: 1

Jean-Christophe Hervé, Nov 16 2013

nonn

Numbers with exactly one prime factor of form 4*k+1, that must have multiplicity one, and no prime factor of the form 4*k+3 with odd multiplicity. There is thus no square in the sequence.
These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.
Numbers such that A004018(a(n)) = 8.
The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.
Also a subsequence of A191217: the two sequences are equal up to a(76) = 320, then A191217(77) = 325, the value which is missing from this sequence, as a(77) = 328 = A191217(78). (3125 is also missing from this sequence, although present in A191217, and it is the 31st such number). - Corrected by Antti Karttunen, May 14 2022.
Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - Altug Alkan, Jul 01 2016
Sequence A125022 (numbers with a unique partition as the sum of 2 squares x^2 + y^2), but without any terms of A028982 (squares and twice squares) that might occur there. - Antti Karttunen, May 14 2022

			a(1) = 5 = 4+1, a(2) = 10 = 9+1, a(3) =  13 = 9+4. However 2 = 1+1, 4 = 4+0, 8 = 4+4 are excluded because the unique decomposition of these numbers in two squares is not with two distinct nonzero squares; 25, 50, 100 are also excluded because there are two decompositions of these numbers in two squares (including one with equal or zero squares).

Antti Karttunen, Table of n, a(n) for n = 1..20000 (extending the previous b-file from Jean-Christophe Hervé, which contained terms up to the 1647th term 10009, but accidentally missed terms 8992 and 9376)
Eric Weisstein's World of Mathematics, Square Number
Wikipedia (fr), Théorème des deux carrés
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).
Subsequence of A004431, of A084645, of A125022, and of A191217.
Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).

isok(n) = {f = factor(n); nb1 = 0; for (i=1, #f~, p = f[i, 1]; ep = f[i, 2]; if (p % 4 == 1, nb1 ++; if (ep != 1, return (0))); if (p % 4 == 3, if (ep % 2, return (0)));); return (nb1 == 1);} \\ Michel Marcus, Nov 17 2013

Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.

{k | A004018(k) = 8}.

A174723 a(n) = n(4n^2 - 3*n + 5)/6.

Original entry on oeis.org

1, 5, 16, 38, 75, 131, 210, 316, 453, 625, 836, 1090, 1391, 1743, 2150, 2616, 3145, 3741, 4408, 5150, 5971, 6875, 7866, 8948, 10125, 11401, 12780, 14266, 15863, 17575, 19406, 21360, 23441, 25653, 28000, 30486, 33115, 35891, 38818, 41900, 45141

Offset: 1

Michel Lagneau, Mar 28 2010

We prove that a(n) = Sum_{k=1..n^2} floor(sqrt(k)): a(n) = Sum_{k=1..3} 1 + Sum_{k=4..8} 2 + ... + Sum_{k=(n-1)^2..n^2 - 1} (n-1) + n = 3*1 + 5*2 + 7*3 + ... + (2n-1)(n-1)+ n = Sum_{k=1..n} (2k-1)*(k-1) + n = 2*Sum_{k=1..n} k^2 - 3*Sum_{k=1..n} k + 2n = 2n(n+1)(2n+1)/6 - 3n(n+1)/2 + 2n = n*(4n^2 - 3n + 5) / 6.

Notice that a(4) = 4 + 3*5 + 2*6 + 1*7 and a(8) = 8 + 7*9 + 6*10 + 5*11 + 4*12 + 3*13 + 2*14 + 1*15. In general, a(n) = n + Sum_{k=1..n-1} (n-k)*(n+k). - J. M. Bergot, Jul 31 2013

			From _Bruno Berselli_, Feb 17 2015: (Start)
Third differences:  1, 2,  4,  4,   4,   4,   4, (repeat 4) ... (A151798)
Second differences: 1, 3,  7, 11,  15,  19,  23,  27,   31, ... (A131098)
First differences:  1, 4, 11, 22,  37,  56,  79, 106,  137, ... (A084849)
-------------------------------------------------------------------------
This sequence:      1, 5, 16, 38,  75, 131, 210, 316,  453, ...
-------------------------------------------------------------------------
Partial sums:       1, 6, 22, 60, 135, 266, 476, 792, 1245, ... (A071239)
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000196, A071239, A084849, A131098, A151798, A022554.

I:=[1, 5, 16, 38]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

A174723 := proc(n)
        n*(4*n^2-3*n+5)/6 ;
end proc:
seq( A174723(n),n=1..20) ; # R. J. Mathar, Nov 07 2011

Table[n (4n^2-3n+5)/6,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,16,38},50] (* Harvey P. Dale, Jan 16 2012 *)

a(n)=n*(4*n^2-3*n+5)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f. x*(1 + x + 2*x^2) / (x-1)^4. - R. J. Mathar, Nov 07 2011
a(1)=1, a(2)=5, a(3)=16, a(4)=38; for n > 4, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jan 16 2012
a(n) = A022554(n^2). - Ridouane Oudra, Jun 13 2025

A032512 Sum of the integer part of 4th roots of integers <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

nonn

Cf. A022554, A031876, A032513.

Partial sums of A255270.

Accumulate[Floor[Surd[Range[0,70],4]]] (* Harvey P. Dale, Dec 14 2024 *)

a(n) = sum(k=1, n, sqrtnint(k, 4)); \\ Michel Marcus, Mar 12 2016

G.f.: Sum_{k>=1} x^(k^4)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016

a(n) = -(1/30) * floor(n^(1/4)) * (-31 - 30 * n + 10 * floor(n^(1/4))^2 + 15 * floor(n^(1/4))^3 + 6 * floor(n^(1/4))^4). - Pooya Farshim, Sep 28 2024

NAME adapted to offset. - Giovanni Resta, May 08 2020

A032513 Sum of the integer part of 5th roots of positive integers less than or equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107

Offset: 0

Michel Tixier (tixier(AT)dyadel.net)

nonn

Cf. A022554, A031876, A032512.

Partial sums of A178487.

Accumulate[Floor[Surd[Range[0,70],5]]] (* Harvey P. Dale, Jan 21 2019 *)

a(n) = sum(k=1, n, sqrtnint(k, 5)); \\ Michel Marcus, Mar 12 2016

G.f.: Sum_{k>=1} x^(k^5)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016

a(n) = -(1/12) * floor(n^(1/5)) * (-12 - 12*n - floor(n^(1/5)) + 5*floor(n^(1/5))^3 + 6*floor(n^(1/5))^4 + 2*floor(n^(1/5))^5). - Pooya Farshim, Sep 28 2024

A174060 a(n) = Sum_{k=0..n} floor(sqrt(k))^2.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 15, 19, 23, 32, 41, 50, 59, 68, 77, 86, 102, 118, 134, 150, 166, 182, 198, 214, 230, 255, 280, 305, 330, 355, 380, 405, 430, 455, 480, 505, 541, 577, 613, 649, 685, 721, 757, 793, 829, 865, 901, 937, 973, 1022, 1071, 1120, 1169, 1218, 1267, 1316

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

Partial sums of A048760. - R. J. Mathar, Mar 31 2010

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Cf. A022554 (1st), this sequence (2nd), A363497 (3rd).

Cf. A363498 (4th), A363499 (5th), A048760.

Accumulate[Table[Floor[Sqrt[k]]^2, {k, 0, 59}]] (* Harvey P. Dale, Jul 13 2013 *)

a(n)=my(m=sqrtint(n+1));(n+1)*m^2-m*(m+1)*(3*m^2+m-1)/6 \\ Charles R Greathouse IV, Jul 04 2013

a(n) = sum(k=0, n, sqrtint(k)^2); \\ Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
A174060 = [0]
for n in range(1,56): A174060.append(A174060[-1] + isqrt(n)**2)
print(A174060) # Karl-Heinz Hofmann, Jun 15 2023

from math import isqrt
def A174060(n): return ((m:=isqrt(n+1))*(6*m*(n+1) - (m+1)*(3*m**2+m-1)))//6
# Karl-Heinz Hofmann, Jun 15 2023

a(n) = (1/6)*m*(6*m*n - (m+1)*(3*m^2+m-1)) with m = floor(sqrt(n)). - Yalcin Aktar, Jan 30 2012

A268173 a(n) = Sum_{k=0..n} (-1)^k*floor(sqrt(k)).

Original entry on oeis.org

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

Offset: 0

John M. Campbell, Jan 28 2016

			a(5) = -1 = floor(sqrt(0)) - floor(sqrt(1)) + floor(sqrt(2)) - floor(sqrt(3)) + floor(sqrt(4)) - floor(sqrt(5)).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

seq(add((-1)^k*floor(sqrt(k)), k=0..n), n=0..80); # Ridouane Oudra, Jan 21 2024

Table[Sum[(-1)^k Floor[Sqrt@ k], {k, 0, n}], {n, 0, 50}] (* Michael De Vlieger, Mar 15 2016 *)

a(n) = sum(k=0, n, (-1)^k*sqrtint(k)); \\ Michel Marcus, Jan 28 2016

a(n) = sqrtint(n)*(-1)^n/2-((-1)^(sqrtint(n)+1)+1)/4; \\ John M. Campbell, Mar 15 2016

a(n) = floor(sqrt(n))*(-1)^n/2 - ((-1)^(floor(sqrt(n))+1)+1)/4.
a(n) = (-1)^n * Sum_{i=1..ceiling(n/2)} c(n+2-2*i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020
From Ridouane Oudra, Jan 21 2024: (Start)
a(n) = (-1)^n*floor((sqrt(n) + (n mod 2))/2);
a(2*n) = floor(sqrt(n/2));
a(2*n+1) = -floor(sqrt((n+1)/2) + 1/2). (End)

Terms a(55) and beyond from Andrew Howroyd, Mar 02 2020

A363497 a(n) = Sum_{k=0..n} floor(sqrt(k))^3.

Original entry on oeis.org

0, 1, 2, 3, 11, 19, 27, 35, 43, 70, 97, 124, 151, 178, 205, 232, 296, 360, 424, 488, 552, 616, 680, 744, 808, 933, 1058, 1183, 1308, 1433, 1558, 1683, 1808, 1933, 2058, 2183, 2399, 2615, 2831, 3047, 3263, 3479, 3695, 3911, 4127, 4343, 4559, 4775, 4991, 5334

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the third powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), this sequence (3rd), A363498 (4th), A363499 (5th).

Table[(n + 1) #^3 - (1/60) # (# + 1) (3 # - 1) (12 #^2 + 7 # - 4) &[Floor@ Sqrt[n]], {n, 0, 50}] (* Michael De Vlieger, Jun 10 2023 *)

a(n) = sum(k=0, n, sqrtint(k)^3); \\ Michel Marcus, Jun 06 2023

from math import isqrt
A363497 = [0]
for n in range(1,50): A363497.append(A363497[-1] + isqrt(n)**3)
print(A363497) # Karl-Heinz Hofmann, Jun 14 2023

from math import isqrt
def A363497(n):return (m:=isqrt(n))**3*(n+1)-(m*(m+1)*(3*m-1)*(12*m**2+7*m-4))//60
# Karl-Heinz Hofmann, Jun 14 2023

a(n) = (n+1)*m^3 - (1/60)*m*(m+1)*(3*m-1)*(12*m^2+7*m-4), where m = floor(sqrt(n)).

A363498 a(n) = Sum_{k=0..n} floor(sqrt(k))^4.

Original entry on oeis.org

0, 1, 2, 3, 19, 35, 51, 67, 83, 164, 245, 326, 407, 488, 569, 650, 906, 1162, 1418, 1674, 1930, 2186, 2442, 2698, 2954, 3579, 4204, 4829, 5454, 6079, 6704, 7329, 7954, 8579, 9204, 9829, 11125, 12421, 13717, 15013, 16309, 17605, 18901, 20197, 21493, 22789

Offset: 0

Hans J. H. Tuenter, Jun 05 2023

Partial sums of the fourth powers of the terms of A000196.

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), this sequence (4th), A363499 (5th).

Table[(n + 1) #^4 - (1/30) # (# + 1)*(20 #^4 + 4 #^3 - 14 #^2 + 4 # + 1) &[Floor@ Sqrt[n]], {n, 0, 45}] (* Michael De Vlieger, Jun 10 2023 *)

from math import isqrt
def A363498(n):
    return (m:=isqrt(n))**4 *(n+1) - (m*(m+1)*(20*m**4+4*m**3-14*m**2+4*m+1))//30
print([A363498(n) for n in range(0,46)]) # Karl-Heinz Hofmann, Jul 15 2023

a(n) = (n+1)*m^4 - (1/30)*m*(m+1)*(20*m^4+4*m^3-14*m^2+4*m+1), where m = floor(sqrt(n)).

cp's OEIS Frontend

A025224 a(n) = floor(Sum_{k=1..n} sqrt(k)).

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A031876 a(n) = Sum_{k=0..n} floor(k^(1/3)).

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A230779 Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

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A174723 a(n) = n*(4*n^2 - 3*n + 5)/6.

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A032512 Sum of the integer part of 4th roots of integers <= n.

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A032513 Sum of the integer part of 5th roots of positive integers less than or equal to n.

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A174060 a(n) = Sum_{k=0..n} floor(sqrt(k))^2.

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A174723 a(n) = n(4n^2 - 3*n + 5)/6.