A343359 -id:A343359 - cp's OEIS frontend

A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

1, 63, 727, 4031, 15559, 45863, 116855, 257983, 526615, 983583, 1755143, 2935231, 4776055, 7407727, 11256623, 16498719, 23859071, 33434063, 46467719, 62949975, 84644439, 111486599, 146142583, 187854119, 240880239, 303814503, 382049919, 473813703, 586746719

Offset: 1

Karl-Heinz Hofmann, May 06 2021

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Joachim von zur Gathen and Jürgen Gerhard, Extract from "3.4. (Non-)Uniqueness of the gcd" chapter, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Column k=6 of A344527.
Cf. A343359, A013664, A018805, A071778, A082540, A082544.
Cf. A342632, A342586, A342935, A342841, A343527, A343193.

a(n)={sum(k=1, n+1, moebius(k)*(n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A343978(n): return sum(mobius(k)*(n//k)**6 for k in range(1, n+1))

from functools import lru_cache
@lru_cache(maxsize=None)
def A343978(n):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A343978(k1)
        j, k1 = j2, n//j2
    return n*(n**5-1)-c+j # Chai Wah Wu, May 17 2021

a(n) = Sum_{k=1..n} mu(k)*floor(n/k)^6.
Lim_{n->infinity} a(n)/n^6 = 1/zeta(6) = A343359 = 945/Pi^6.
a(n) = n^6 - Sum_{k=2..n} a(floor(n/k)). - Seiichi Manyama, Sep 13 2024

Edited by N. J. A. Sloane, Jun 13 2021

A344038 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= 10^n).

Original entry on oeis.org

1, 983583, 983029267047, 982960635742968103, 982953384128772770413831, 982952672223441253533233827367, 982952600027678075050509511271466303, 982952593055042000417993486008754893529583, 982952592342881094406730790044111038427637071855

Offset: 0

Karl-Heinz Hofmann, May 07 2021

Chai Wah Wu, Table of n, a(n) for n = 0..15

Cf. A342586, A342841, A343193, A343282, A343359, A013664.
Cf. A343978, A071778, A018805, A082540, A082544.
Related counts of k-tuples:
pairs: A018805, A342632, A342586;
triples: A071778, A342935, A342841;
quadruples: A082540, A343527, A343193;
5-tuples: A343282;
6-tuples: A343978, A344038. - N. J. A. Sloane, Jun 13 2021

a(n)={sum(k=1, 10^n+1, moebius(k)*(10^n\k)^6)} \\ Andrew Howroyd, May 08 2021

from labmath import mobius
def A344038(n): return sum(mobius(k)*(10**n//k)**6 for k in range(1, 10**n+1))

Lim_{n->infinity} a(n)/10^(6*n) = 1/zeta(6) = A343359 = 945/Pi^4.

a(n) = A343978(10^n).

Edited by N. J. A. Sloane, Jun 13 2021

A342683 Decimal expansion of 1/zeta(8).

Original entry on oeis.org

9, 9, 5, 9, 3, 9, 2, 0, 1, 1, 2, 5, 5, 1, 5, 1, 4, 6, 8, 3, 4, 8, 3, 6, 4, 7, 2, 8, 0, 5, 5, 4, 5, 3, 2, 4, 0, 0, 5, 0, 2, 2, 7, 7, 8, 4, 5, 8, 9, 3, 0, 3, 6, 2, 7, 8, 5, 3, 5, 4, 2, 4, 5, 5, 5, 4, 1, 3, 8, 5, 7, 4, 6, 2, 0, 9, 4, 0, 4, 5, 4, 2, 6, 5, 1, 5, 9

Offset: 0

Karl-Heinz Hofmann, May 18 2021

1/zeta(8) is the probability that 8 randomly selected numbers will be coprime.

			0.9959392011255151468348364728055453240050227784589...

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
Wikipedia, Riemann zeta function.
Index entries for transcendental numbers

Cf. A008683, A059956, A215267, A343359, A013666.

evalf(9450/Pi^8,100) ; # R. J. Mathar, Jun 04 2021

RealDigits[1/Zeta[8], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

PARI
```
1/zeta(8)
```

Equals 1/A013666 = 9450/Pi^8.
From Amiram Eldar, Jun 01 2023: (Start)
Equals Sum_{k>=1} mu(k)/k^8, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^8). (End)

A355265 Bicubeful numbers.

Original entry on oeis.org

64, 128, 192, 256, 320, 384, 448, 512, 576, 640, 704, 729, 768, 832, 896, 960, 1024, 1088, 1152, 1216, 1280, 1344, 1408, 1458, 1472, 1536, 1600, 1664, 1728, 1792, 1856, 1920, 1984, 2048, 2112, 2176, 2187, 2240, 2304, 2368, 2432, 2496, 2560, 2624, 2688, 2752

Offset: 1

Peter Luschny, Jul 12 2022

nonn

Let lp(n, e) denote the largest positive integer b such that b^e divides n. For example for e = 1, 2, 3, 4 the sequences (lp(n, e), n >= 1) are A000027, A000188, A053150, and A053164. Let rad(n) = A007947(n) be the squarefree kernel of n. k is in this sequence if lp(n, 3) does not divide rad(n). The case e = 1 gives A013929, and the case e = 2 is A046101.

The asymptotic density of this sequence is 1 - 1/zeta(6) = 1 - 945/Pi^6 = 0.017047... . - Amiram Eldar, Jul 13 2022

			n = 512 = 2^9, rad(n) = 2, lp(n, 3) = 8 since n/8^3 = 1. But 8 does not divide 2.
n = 704 = 2^6*11, rad(n) = 22, lp(n, 3) = 4 since n/4^3 = 11. But 4 does not divide 22.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007947, A000188, A053150, A053164, A013929, A046101 (biquadrateful).

Cf. A013664, A343359.

with(NumberTheory):
isBicubeful := n -> irem(Radical(n), LargestNthPower(n, 3)) <> 0:
select(isBicubeful, [`$`(1..2752)]);

bicubQ[n_] := AnyTrue[FactorInteger[n][[;; , 2]], # > 5 &]; Select[Range[3000], bicubQ] (* Amiram Eldar, Jul 13 2022 *)

from itertools import count, islice
from sympy import factorint
def A355265_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(map(lambda m:m>5,factorint(n).values())),count(max(startvalue,1)))
A355265_list = list(islice(A355265_gen(),30)) # Chai Wah Wu, Jul 12 2022

A number k is bicubeful iff it is divisible by the 6th power of an integer > 1.

A341901 Decimal expansion of 1/zeta(9).

Original entry on oeis.org

9, 9, 7, 9, 9, 5, 6, 3, 2, 7, 3, 0, 7, 6, 2, 1, 5, 6, 8, 6, 4, 6, 7, 6, 1, 3, 2, 1, 0, 5, 0, 9, 9, 9, 9, 6, 3, 2, 0, 9, 4, 1, 8, 4, 8, 0, 5, 1, 8, 2, 1, 1, 9, 1, 2, 3, 7, 3, 6, 7, 4, 5, 1, 3, 3, 7, 5, 2, 3, 0, 1, 0, 5, 1, 9, 4, 1, 1, 4, 1, 8, 2, 4, 3, 9, 1, 7

Offset: 0

Karl-Heinz Hofmann, Jun 04 2021

1/zeta(9) is the probability that 9 randomly selected numbers will be coprime.

			0.997995632730762156864676132105099996320941848...

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

Cf. A008683, A013667, A059956, A088453, A215267, A343308, A343359, A343367, A342683.

Mathematica
```
RealDigits[1/Zeta[9], 10, 100][[1]]
```
PARI
```
1/zeta(9)
```

Equals 1/A013667.
From Amiram Eldar, Jun 01 2023: (Start)
Equals Sum_{k>=1} mu(k)/k^9, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^9). (End)

cp's OEIS Frontend

A343978 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= n).

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A344038 Number of ordered 6-tuples (a,b,c,d,e,f) with gcd(a,b,c,d,e,f)=1 (1<= {a,b,c,d,e,f} <= 10^n).

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A342683 Decimal expansion of 1/zeta(8).

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A355265 Bicubeful numbers.

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A341901 Decimal expansion of 1/zeta(9).

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