A031876 -id:A031876 - cp's OEIS frontend

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

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

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

A270370 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/3)).

Original entry on oeis.org

0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 2, -2, 2, -2, 2, -2

Offset: 0

John M. Campbell, Mar 15 2016

			a(5) = [0^(1/3)]-[1^(1/3)]+[2^(1/3)]-[3^(1/3)]+[4^(1/3)]-[5^(1/3)] = 0-1+1-1+1-1 = -1, letting [] denote the floor function.

Cf. A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^i*Floor[i^(1/3)],{i,0,n}],{n,0,100}]]

PARI
```
a(n)=sum(i=0,n,(-1)^i*sqrtnint(i,3))
```

a(n)=sqrtnint(n,3)*(-1)^n/2-((-1)^(sqrtnint(n,3)+1)+1)/4

a(n) = floor(n^(1/3))*(-1)^n/2 - ((-1)^(floor(n^(1/3))+1)+1)/4.

A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 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, 139, 143, 147, 151, 155, 159, 163, 167

Offset: 1

Yi Yang, Sep 20 2017

a(n) > c*n*log(n)*log(log(n))*log(log(log(n)))*...*log(log...(log(n))...) (k layers) for any sufficient large n, any constant c and any positive integer k.

The sum of 1/a(i) for i = 1, 2, 3, ... diverges extremely slowly.

Robert Israel, Table of n, a(n) for n = 1..10000
KeyTo9_Fans, A Chinese post discussing the sum of 1/a(i)
Fedor Petrov, The proof of the divergence of the sum of 1/a(i), Mathoverflow, Sep 2017.

Cf. A000195, A031876, A292620.

f:= proc(n) option remember;
procname(n-1)+procname(floor(log(n)))
end proc:
f(1):= 1: f(2):= 2:
map(f, [$1..100]); # Robert Israel, Sep 28 2017

a[n_] := a[n] = If[n <= 2, n, a[n - 1] + a[Floor@ Log@ n]]; Array[a, 62] (* Michael De Vlieger, Sep 21 2017 *)

a(n) = if (n<=2, n, a(n-1) + a(floor(log(n)))); \\ Michel Marcus, Sep 21 2017

A262352 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/4)).

Original entry on oeis.org

0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1, -2, 1

Offset: 0

John M. Campbell, Mar 24 2016

			Letting [] denote the floor function, a(7) = [0^(1/4)] - [1^(1/4)] + [2^(1/4)] - [3^(1/4)] + [4^(1/4)] - [5^(1/4)] + [6^(1/4)] - [7^(1/4)] = 0 - 1 + 1 - 1 + 1 - 1 + 1 - 1 = -1.

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A270370, A268173, A022554, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^k*Floor[k^(1/4)],{k,0,n}],{n,0,100}]] ;

a(n)=floor(n^(1/4))*(-1)^n/2-((-1)^(floor(n^(1/4))+1)+1)/4

PARI
```
a(n)=sum(k=0,n,(-1)^k*floor(k^(1/4)))
```

A262352(n) = sum(k=0,n,((-1)^k)*sqrtnint(k, 4)); \\ Antti Karttunen, Nov 06 2018

a(n) = floor(n^(1/4))*(-1)^n/2-((-1)^(floor(n^(1/4))+1)+1)/4.

More terms from Antti Karttunen, Nov 06 2018

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

A270103 Array read by antidiagonals: T(n, k) is the sum of the integer part of the n-th roots of natural numbers less than k.

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 3, 2, 1, 15, 5, 3, 2, 1, 21, 7, 4, 3, 2, 1, 28, 9, 5, 4, 3, 2, 1, 36, 11, 6, 5, 4, 3, 2, 1, 45, 13, 7, 6, 5, 4, 3, 2, 1, 55, 16, 9, 7, 6, 5, 4, 3, 2, 1, 66, 19, 11, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

John M. Campbell, Mar 11 2016

			The fifth entry in the second row of this array is 7, since 7 = floor(sqrt(1)) + floor(sqrt(2)) + floor(sqrt(3)) + floor(sqrt(4)) + floor(sqrt(5)).
The table array begins:
  1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
  1, 2, 3,  5,  7,  9, 11, 13, 16, 19, ...
  1, 2, 3,  4,  5,  6,  7,  9, 11, 13, ...
  1, 2, 3,  4,  5,  6,  7,  8,  9, 10, ...
  ...

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.

M. Griffiths, More Sums Involving the Floor Function, Math. Gaz., 86 (2002), 285-287.
Maths StackExchange, Find a formula for Sum_{k=1..n} floor(sqrt(k)), see achille hui formula.
Wikipedia, Faulhaber's formula

Rows 1 to 5: A000217, A022554, A031876, A032512, A032513.

Main diagonal and each diagonal below the main diagonal: A000027.

T[n_, k_] := (1 + k) Floor[k^(1/n)] - HarmonicNumber[Floor[k^(1/n)], -n] (* Daniel Hoying, Jun 11 2020 *)

T(n, k) = sum(j=0, k, sqrtnint(j, n)); \\ Michel Marcus, Mar 12 2016

T(n,k) = Sum_{j=0..k} floor(j^(1/n)).
T(n,k) = (1+k)*floor(k^(1/n)) - (1/(n+1))*Sum_{j=1..n+1} (1 + floor(k^(1/n)))^j*binomial(n+1, j)*Bernoulli(n+1-j).
T(n,k) = (1+k)*floor(k^(1/n)) - Sum_{j=1..floor(k^(1/n))} j^n. - Daniel Hoying, Jun 11 2020

A270825 a(n) = Sum_{i=0..n} (-1)^floor(i/2)*floor(sqrt(i)).

Original entry on oeis.org

0, 1, 0, -1, 1, 3, 1, -1, 1, 4, 1, -2, 1, 4, 1, -2, 2, 6, 2, -2, 2, 6, 2, -2, 2, 7, 2, -3, 2, 7, 2, -3, 2, 7, 2, -3, 3, 9, 3, -3, 3, 9, 3, -3, 3, 9, 3, -3, 3, 10, 3, -4, 3, 10, 3, -4, 3, 10, 3, -4, 3, 10, 3, -4, 4, 12, 4, -4, 4, 12, 4, -4, 4, 12, 4, -4, 4

Offset: 0

John M. Campbell, Mar 23 2016

			Letting [] denote the floor function, a(7) = [sqrt(0)]+[sqrt(1)]-[sqrt(2)]-[sqrt(3)]+[sqrt(4)]+[sqrt(5)]-[sqrt(6)]-[sqrt(7)] = 0+1-1-1+2+2-2-2 = -1.

Cf. A268173, A022554, A270370, A031876, A032512, A032513, A032514, A032515, A032516, A032517, A032518, A032519, A032520, A032521.

Print[Table[Sum[(-1)^(Floor[i/2])*Floor[Sqrt[i]],{i,0,n}],{n,0,100}]]

a(n)=sum(i=0,n,(-1)^(floor(i/2))*floor(sqrt(i)))

a(4m)=floor(sqrt(m)), a(4m+1)=floor(3/2*floor(sqrt(4m+1))), a(4m+2)=floor(sqrt(m)), a(4m+3)=-floor((1+sqrt(4m+3))/2).

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A270370 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/3)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A262352 a(n) = Sum_{k=0..n} (-1)^k*floor(k^(1/4)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI