A255270 -id:A255270 - cp's OEIS frontend

A178487 a(n) = floor(n^(1/5)): integer part of fifth root of n.

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

Offset: 0

M. F. Hasler, Oct 09 2010

Each term k appears (k+1)^5 - k^5 times consecutively (A022521). - Bernard Schott, Mar 07 2023

Sequences a(n) = floor(n^(1/k)): A001477 (k=1), A000196 (k=2), A048766 (k=3), A255270 (k=4), this sequence (k= 5), A178489 (k=6), A057427 (k->oo).

Cf. A022521, A253206.

[n eq 0 select 0 else Iroot(n, 5): n in [0..110]]; // Bruno Berselli, Feb 20 2015

seq(floor(n^(1/5)), n=0..100); # Ridouane Oudra, Feb 26 2023

Floor[Range[0,120]^(1/5)] (* Harvey P. Dale, Aug 15 2012 *)

PARI
```
A178487(n)=floor(sqrtn(n+.5,5))
```

a(n) = sqrtnint(n, 5); \\ Michel Marcus, Dec 22 2016

from sympy import integer_nthroot
def A178487(n): return integer_nthroot(n,5)[0] # Chai Wah Wu, Jun 06 2025

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

a(n) = Sum_{i=1..n} A253206(i)*floor(n/i). - Ridouane Oudra, Feb 26 2023

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

A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

Original entry on oeis.org

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

Offset: 1

V. Raman, Sep 03 2012

nonn

			From _Antti Karttunen_, Aug 28 2017: (Start)
For n = 2 there is one solution: 2 = 1^4 + 1^4, thus a(2) = 1.
For n = 17 there is one solution: 17 = 2^4 + 1^4, thus a(17) = 1.
For n = 635318657 we have two solutions: 635318657 = 158^4 + 59^4 = 134^4 + 133^4, thus a(635318657) = 2. Note that this is the first point where the sequence attains value greater than 1. See _Charles R Greathouse IV_'s Jan 12 2017 comment in A216280.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A025455, A025446, A000161, A025426, A216280.

(define (A216284 n) (let loop ((x (A255270 n)) (s 0)) (let* ((x4 (A000583 x)) (y4 (- n x4))) (if (< x4 y4) s (loop (- x 1) (+ s (if (and (> y4 0) (= (A000583 (A255270 y4)) y4)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

a(n) <= A216280(n). - Antti Karttunen, Aug 28 2017

Definition edited to match the given data and the second part of offset (635318657) explicitly added by Antti Karttunen, Aug 28 2017

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).

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.

A339276 Nearest integer to the fourth root of n.

Original entry on oeis.org

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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Amiram Eldar, Dec 13 2020

			a(1) = 1 since 1^(1/4) = 1.
a(6) = 2 since 6^(1/4) = 1.565... and its nearest integer is 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein and others, Nearest Integer Zeta Functions, solution to Problem 10212, The American Mathematical Monthly, Vol. 101, No. 6 (1994), pp. 579-580.

Cf. A000194, A105209, A255270.

Mathematica
```
Table[Round[Surd[n, 4]], {n, 1, 100}]
```

from sympy import integer_nthroot
def A339276(n): return (m:=integer_nthroot(n,4)[0])+((n<<4)>=((m<<1)+1)**4) # Chai Wah Wu, Jun 06 2025

Sum_{n>=1} 1/a(n)^s = 4*zeta(s-3) + zeta(s-1), for s>4 (Borwein, 1994).

A373652 Composite numbers k for which g = gcd(f(i*c), k) = 1 or k for all i in the range 1 <= i <= c, where f(x) = Product_{j=1..c} x+j and c = floor(k^(1/4)).

Original entry on oeis.org

9, 15, 49, 77, 169, 221, 247, 323, 529, 961, 1147, 1271, 1517, 1680, 1849, 2021, 2209, 2279, 2520, 2688, 2880, 3360, 3481, 3599, 3721, 3953, 4032, 4087, 4320, 4480, 4536, 4757, 5040, 5184, 5329, 5670, 5760, 5767, 6048, 6059, 6241, 6480, 6497, 6557, 6720, 7200

Offset: 1

Darío Clavijo, Jun 12 2024

nonn

These conditions for k are inspired by the Pollard-Strassen factorization algorithm.
f(i*c) is the product of successive blocks of consecutive integers c*i+1 to c*(i+1) inclusive and can be calculated efficiently mod k by a multi-point polynomial evaluation.
The conditions here are that no g by itself reveals a factor of k (so that the Pollard-Strassen algorithm must examine individual c*i+j terms in some g = k block to find a factor).

			For k = 49:
 i | c |  c*i+1 | c*(i+1) | g
-------------------------------------
 1 | 2 | 3      | 4       | 1
 2 | 2 | 5      | 6       | 1
Result: 1
For k = 323:
 i | c | c*i+1 | c*(i+1) | g
-------------------------------------
 1 | 4 | 5     | 8       | 1
 2 | 4 | 9     | 12      | 1
 3 | 4 | 13    | 16      | 1
 4 | 4 | 17    | 20      | 323
Result: 323

Mathematics Stack Exchange, Pollard-Strassen algorithm.
Programming Praxis, Strassen's factoring algorithm.

Cf. A002808, A255270.

s(n) = my(c=sqrtnint(n, 4), vf = vector(c, k, 1)); for (i=1, #vf, vf[i] = prod(j=c*i+1, c*(i+1), j % n); vf[i] = gcd(vf[i], n);); vf;
isok(n) = if ((n>1) && !isprime(n), my(x=Set(s(n)), y=Set([1,n])); setunion(x, y) == y); \\ Michel Marcus, Jun 18 2024

from sympy import integer_nthroot, gcd, isprime
def s(k):
  c = integer_nthroot(k, 4)[0]
  f = [1]*c
  for i in range(1, c+1):
    for j in range(c*i+1, c*(i+1)+1):
      f[i-1] = (f[i-1] * j) % k
    f[i-1] = gcd(f[i-1], k)
  return f
isok = lambda k: not isprime(k) and not any(k > x > 1 for x in s(k))
print([k for k in range(4, 7200) if isok(k)])

from itertools import count, islice
from math import gcd, prod
from sympy import isprime
def A373652_gen(): # generator of terms
    for c in count(1):
        g = [prod(i*c+j for j in range(1,c+1)) for i in range(1,c+1)]
        yield from filter(lambda k: not (k==1 or isprime(k) or any(1A373652_list = list(islice(A373652_gen(),20)) # Chai Wah Wu, Jul 16 2024

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A178487 a(n) = floor(n^(1/5)): integer part of fifth root of n.

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

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A216284 Number of solutions to the equation x^4+y^4 = n with x >= y > 0.

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

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

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A339276 Nearest integer to the fourth root of n.

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A373652 Composite numbers k for which g = gcd(f(i*c), k) = 1 or k for all i in the range 1 <= i <= c, where f(x) = Product_{j=1..c} x+j and c = floor(k^(1/4)).

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