A215267 -id:A215267 - cp's OEIS frontend

A343359 Decimal expansion of 1/zeta(6).

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

Offset: 0

Karl-Heinz Hofmann, Apr 12 2021

Decimal expansion of 1/zeta(6), the inverse of A013664.
1/zeta(6) has a known closed-form formula (945/Pi^6) like 1/zeta(2) = 6/Pi^2 and 1/zeta(4) = 90/Pi^4.
1/zeta(6) is the probability that 6 randomly selected numbers will be coprime. - A.H.M. Smeets, Apr 13 2021

			0.982952592264580419804896564993924132951221515986...

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

Cf. A008683, A059956, A088453, A215267, A343308.

RealDigits[1/Zeta[6], 10, 100][[1]] (* Amiram Eldar, Apr 12 2021 *)

PARI
```
1/zeta(6) \\ A.H.M. Smeets, Apr 13 2021
```

Equals 1/A013664 = 945/Pi^6.
From Amiram Eldar, Jun 01 2023: (Start)
Equals Sum_{k>=1} mu(k)/k^6, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^6). (End)

A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600

Offset: 1

Amiram Eldar, Jul 29 2024

Subsequence of A176297 and first differs from it at n = 41: A176297(41) = 432 = 2^4 * 3^3 is not a term of this sequence.
Numbers whose prime factorization has least one exponent that equals 3 and no higher exponent.
Numbers k such that A051903(k) = 3.
The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) = A215267 - A088453 = 0.0920310303408826983406... .

Intersection of A046100 and A176297.
Cf. A004709, A051903, A067259.
Cf. A002117, A013662, A088453, A215267.

Select[Range[600], Max[FactorInteger[#][[;; , 2]]] == 3 &]

is(k) = k > 1 && vecmax(factor(k)[,2]) == 3;

from sympy import mobius, integer_nthroot
def A375072(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**4-x//k**3) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**3) for k in range(integer_nthroot(x,4)[0]+1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A375245 Number of biquadratefree numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 75, 75, 76, 77, 78, 79

Offset: 1

Chai Wah Wu, Aug 07 2024

First differs from A309083 at n = 81: a(81) = 75, A309083(n) = 77. - Andrew Howroyd, Aug 10 2024

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A060431, A013662, A046100, A215267, A307430.

Accumulate[Table[Boole[Max[FactorInteger[n][[;; , 2]]] < 4], {n, 1, 100}]] (* Amiram Eldar, Aug 10 2024 *)

a(n) = sum(d=1, sqrtnint(n,4), moebius(d)*(n\d^4)) \\ Andrew Howroyd, Aug 10 2024

from sympy import mobius, integer_nthroot
def A375245(n): return int(sum(mobius(k)*(n//k**4) for k in range(1, integer_nthroot(n,4)[0]+1)))

a(n) = Sum_{d>=1} mu(d)*floor(n/d^4), where mu is the Moebius function A008683.
n/a(n) converges to zeta(4).
a(n) = Sum_{k = 1..n} A307430(k).

a(68) onwards from Andrew Howroyd, Aug 10 2024

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)

A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

Original entry on oeis.org

72, 108, 200, 360, 392, 500, 504, 540, 600, 675, 756, 792, 936, 968, 1125, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1500, 1656, 1800, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2312, 2484, 2520, 2600, 2646, 2664, 2700, 2888, 2904, 2952, 3087

Offset: 1

Amiram Eldar, Jul 29 2024

Numbers whose powerful part (A057521) is a term of A375073.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) + 1/zeta(2) - zeta(6)/(zeta(2) * zeta(3)) * c = A215267 - A088453 + A059956 - A068468 * c = 0.0156712080080470088619..., where c = Product_{p prime} (1 + 2/p^3 - 1/p^4 + 1/p^5).

Equals A046100 \ (A004709 UNION A336591).
Disjoint union of A375073 and A375075.
Cf. A051903, A057521.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A215267.

Select[Range[3000], Union[Select[FactorInteger[#][[;; , 2]], # > 1 &]] == {2, 3} &]

is(k) = Set(select(x -> x > 1, factor(k)[,2])) == [2, 3];

A051903(a(n)) = 3.

A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1500, 1656, 1836, 1960, 2052, 2088, 2200, 2232, 2250, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3096, 3132, 3348, 3384, 3400, 3500, 3780, 3800, 3816, 3960, 3996, 4056, 4116, 4200, 4248, 4312, 4392, 4428

Offset: 1

Amiram Eldar, Jul 29 2024

First differs from its subsequence A163569 at n = 25: a(25) = 2520 = 2^3 * 3^2 * 5 * 7 is not a term of A163569.
Numbers k such that the set of distinct prime factorization exponents of k (row k of A136568) is {1, 2, 3}.
The asymptotic densities of this sequence and A375074 are equal (0.0156712..., see A375074 for a formula), since the terms in A375074 that are not in this sequence (A375073) have a density 0.

Intersection of A375072 and A317090.
Equals A375074 \ A375073.
Subsequence of A046100 and A176297.
A163569 is a subsequence.
Cf. A002117, A013661, A013662, A013664, A059956, A068468, A088453, A136568, A215267.

Select[Range[4500], Union[FactorInteger[#][[;; , 2]]] == {1, 2, 3} &]

is(k) = Set(factor(k)[,2]) == [1, 2, 3];

A375246 Number of biquadratefree numbers <= 10^n.

Original entry on oeis.org

1, 10, 93, 925, 9240, 92395, 923939, 9239385, 92393839, 923938406, 9239384029, 92393840300, 923938402926, 9239384029211, 92393840292169, 923938402921591, 9239384029215891, 92393840292159004, 923938402921590127, 9239384029215901651, 92393840292159016603

Offset: 0

Chai Wah Wu, Aug 07 2024

nonn

Digits of terms converge to digits of 1/zeta(4) = 90/Pi^4 (A215267).

Chai Wah Wu, Table of n, a(n) for n = 0..39

Cf. A060431, A013662, A046100, A215267, A307430, A375245.

from sympy import mobius, integer_nthroot
def A375246(n): return int(sum(mobius(k)*(10**n//k**4) for k in range(1, integer_nthroot(10**n,4)[0]+1)))

a(n) = A375245(10^n).

A382967 Biquadratefree numbers (A046100) that are not squarefree (A005117).

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 84, 88, 90, 92, 98, 99, 100, 104, 108, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 180, 184

Offset: 1

Amiram Eldar, Apr 10 2025

Subsequence of A252849 and first differs from it at n = 22: A252849(22) = 64 = 2^6 is not a term of this sequence.
Subsequence of A375229 and differs from it by not having the terms 1, 256, 512, 768, 1024, ... .
Numbers whose prime factorization has least one exponent that equals 2 or 3 and no higher exponent.
Numbers k such that 2 <= A051903(k) <= 3.
The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(2) = A215267 - A059956 = 0.3160113... .

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

Disjoint union of A067259 and A375072.
Intersection of A046100 and A013929.
Subsequence of A252849 and A375229.
Cf. A004709, A005117, A051903, A059956, A215267.

Select[Range[200], 2 <= Max[FactorInteger[#][[;; , 2]]] <= 3 &]

isok(k) = if(k == 1, 0, my(emax = vecmax(factor(k)[, 2])); emax > 1 & emax < 4);

from math import isqrt
from sympy import mobius, integer_nthroot
def A382967(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+sum(mobius(k)*(x//k**2-x//k**4) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,4)[0]+1,isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Apr 11 2025

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (12*(15 - Pi^2)). - Vaclav Kotesovec, Apr 11 2025

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)

A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 31 2024

The asymptotic mean of the ratio between the number of non-isomorphic abelian groups and the number of non-isomorphic semisimple rings of the same order.
The constant A in Kühleitner's paper (1995).
The ratio is 1 for all biquadratefree numbers (whose asymptotic density is A215267 = 0.923..., see A046100), and smaller than 1 otherwise.

			0.98771484004493763774023068670639349351901075670395...

Manfred Kühleitner, Comparing the number of Abelian groups and of semisimple rings of a given order, Mathematica Slovaca, Vol. 45, No. 5 (1995), pp. 509-518.

Cf. A000041, A000688, A004101, A038538, A046100, A215267.

default(realprecision, 120); my(N=512, x='x+O('x^N), v); v = Vec(1/prod(k=1, sqrtint(N)+1, prod(j=1, 1+N\k^2, 1-x^(j*k^2)))); prodeulerrat((1-1/p)*vecsum(vector(N, i, numbpart(i-1)/(v[i]*p^(i-1))))) \\ after Vaclav Kotesovec at A004101

Equals Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} A000041(k)/(A004101(k)*p^k)).

cp's OEIS Frontend

A343359 Decimal expansion of 1/zeta(6).

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A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

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A375245 Number of biquadratefree numbers <= n.

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A342683 Decimal expansion of 1/zeta(8).

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A375074 Numbers whose prime factorization exponents include at least one 2, at least one 3 and no higher exponents.

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A375075 Numbers whose prime factorization exponents include at least one 1, at least one 2, at least one 3 and no other exponents.

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A375246 Number of biquadratefree numbers <= 10^n.

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