A178487 -id:A178487 - cp's OEIS frontend

A255270 Integer part of fourth root of n.

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 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

Offset: 0

Bruno Berselli, Feb 20 2015

n appears (n+1)^4 - n^4 times (A005917).

Bruno Berselli, Table of n, a(n) for n = 0..1000

Cf. A005917.
Cf. sequences of the type floor(n^(1/k)): A000196 (k=2), A048766 (k=3), this sequence (k=4), A178487 (k=5), A178489 (k=6).
Cf. A219009.

[IsZero(n) select 0 else Iroot(n, 4): n in [0..100]];

[Floor(n^(1/4)): n in [0..100]]; // Vincenzo Librandi, Feb 20 2015

A255270 := proc(n)
    floor( n^(1/4)) ;
end proc:
seq(A255270(n),n=0..100) ; # R. J. Mathar, May 08 2020

Mathematica
```
Floor[Range[0, 100]^(1/4)]
```
PARI
```
vector(100, n, n--; floor(n^(1/4)))
```

a(n) = sqrtnint(n, 4); \\ Michel Marcus, Dec 22 2016

from sympy import integer_nthroot
def A255270(n): return integer_nthroot(n,4)[0] # Chai Wah Wu, Jun 06 2025

Sage
```
[floor(n^(1/4)) for n in (0..100)]
```

a(n) = floor(n^(1/4)) = floor(sqrt(A000196(n))).
G.f.: Sum_{k>=1} x^(k^4)/(1 - x). - Ilya Gutkovskiy, Dec 22 2016
a(n) = Sum_{i=1..n} A219009(i)*floor(n/i). - Ridouane Oudra, Feb 26 2023

A178489 a(n) = floor(n^(1/6)): integer part of sixth root of n.

Original entry on oeis.org

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, 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, 2, 2, 2, 2, 2, 2, 2

Offset: 0

M. F. Hasler, Oct 09 2010

Cf. A000196, A048766, A178487.

Floor[Power[Range[0,110], (6)^-1]] (* Harvey P. Dale, Jul 18 2011 *)

PARI
```
A178489(n)=floor(sqrtn(n+.5,6))
```

a(n) = sqrtnint(n, 6); \\ Michel Marcus, Dec 22 2016

G.f.: Sum_{k>=1} x^(k^6)/(1 - x). - Ilya Gutkovskiy, Dec 22 2016

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

A381042 Alternating sum of floor(n^(1/k)), with k >= 2.

Original entry on oeis.org

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

Offset: 0

Friedjof Tellkamp, Apr 14 2025

nonn

			n:       0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
k=2 (+): 0, 1, 1, 1, 2, 2, 2, 2, 2, 3, ... (A000196)
k=3 (-): 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, ... (A048766)
...
Sum:     0, 0, 0, 0, 1, 1, 1, 1, 0, 1 ... (= this sequence).

Friedjof Tellkamp, Table of n, a(n) for n = 0..10000

Cf. A000196 (k=2), A048766 (k=3), A255270 (k=4), A178487 (k=5), A178489 (k=6).

Cf. A089361 (nonalternating), A382691, A382692.

z = 100; Table[Sum[(-1)^k Floor[n^(1/k)], {k, 2, 2 Floor@Log[2, z/2] - 1}], {n, 0, z}]

a(n) = A000196(n) - A048766(n) + A255270(n) - A178487(n) + ... .
a(n) = Sum_{k>=2} (-1)^k * floor(n^(1/k)) = Sum_{k>=1} (floor(n^(1/(2*k))) - floor(n^(1/(2*k+1)))).
a(n) = Sum_{i=1..n} A382691(i).
a(n) ~ A382692(n).
G.f.: Sum_{j>=1, k>=2} (-1)^k * x^(j^k)/(1-x).

A132336 Sum of the integers from 1 to n, excluding perfect fifth powers.

Original entry on oeis.org

0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 495, 528, 562, 597, 633, 670, 708, 747, 787, 828, 870, 913, 957, 1002, 1048, 1095, 1143, 1192, 1242, 1293, 1345, 1398, 1452

Offset: 1

Cino Hilliard, Nov 07 2007

			a(1)=0+1, excluding 0 and 1, so a(1)=0.
a(2)=0+1+2, excluding 0 and 1, so a(2)=2.
a(3)=0+1+2+3, excluding 0 and 1, so a(3)=2+3=5.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000217, A000539, A178487.
Different from A000096.
Cf. A132337.

A000217 := proc(n) n*(n+1)/2 ; end proc:
A000539 := proc(n) (2*n^6+6*n^5+5*n^4-n^2)/12 ; end proc:
A132336 := proc(n) r := floor(n^(1/5)) ; A000217(n)-A000539(r); end proc: seq(A132336(n),n=1..40) ;

g5(n)=for(x=1, n, r=floor(x^(1/5)); sum5=(2*r^6+6*r^5+5*r^4-r^2)/12; sn=x* (x+1)/2; print1(sn-sum5, ", "))

a(n) = my(r=sqrtnint(n,5)); n*(n+1)/2 - (2*r^6+6*r^5+5*r^4-r^2)/12; \\ Ruud H.G. van Tol, Nov 02 2023

from sympy import integer_nthroot
def A132336(n): return n*(n+1)-(m:=integer_nthroot(n,5)[0])**2*(m**2*(m*(m+3<<1)+5)-1)//6>>1 # Chai Wah Wu, Jun 06 2025

a(n) = A000217(n) - A000539(r) where r = floor(n^(1/5)).
a(n) = n(n+1)/2 - (2r^6 + 6r^5 + 5r^4 - r^2)/12.
a(n) = A000217(n) - A000539(r) where r= A178487(n). - R. J. Mathar, Oct 12 2010

Edited by the Assoc. Editors of the OEIS, Oct 12 2010. Thanks to Daniel Mondot for pointing out that the sequence needed editing.

A382692 a(n) = floor of alternating sum of k-th roots of n, with k >= 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6

Offset: 0

Friedjof Tellkamp, Apr 05 2025

nonn

			a(15) = floor(15^(1/2) - 15^(1/3) + 15^(1/4) - ...) = floor(1.9701...) = 1.

Cf. A000196 (k=2), A048766 (k=3), A255270 (k=4), A178487 (k=5), A178489 (k=6).

Cf. A381042, A382691.

Floor@Table[NSum[n^(1/(2 k)) - n^(1/(2 k + 1)), {k, 1, Infinity}, WorkingPrecision -> 30], {n, 1, 100}]

a(n) = floor(Sum_{k>=2} (-1)^k * n^(1/k)) = floor(Sum_{k>=1} (n^(1/(2*k)) - n^(1/(2*k + 1)))).
a(n) ~ A381042(n).
Series expansion of Sum_{k>=2} (-1)^k * x^(1/k) at x=1: Sum_{i>=0} A(i) * (x-1)^i/i!, where A(i) = KroneckerDelta(i, 1) - Sum_{j=1..i} eta(j) * StirlingS1(i, j), with eta as the Dirichlet eta function.

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A255270 Integer part of fourth root of n.

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A178489 a(n) = floor(n^(1/6)): integer part of sixth root of 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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A381042 Alternating sum of floor(n^(1/k)), with k >= 2.

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A132336 Sum of the integers from 1 to n, excluding perfect fifth powers.

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A382692 a(n) = floor of alternating sum of k-th roots of n, with k >= 2.

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