A082020 -id:A082020 - cp's OEIS frontend

A383057 Decimal expansion of the asymptotic mean of A056671(k)/A034444(k), the ratio between the number of squarefree unitary divisors and the number of unitary divisors over the positive integers.

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

Offset: 0

Amiram Eldar, Apr 15 2025

The asymptotic mean of the inverse ratio A034444(k)/A056671(k) is 15/Pi^2 (A082020).

			0.78936260126098902910370862925139689276851676052691...

The unitary analog of A308043.

Cf. A034444, A056671, A082020, A383058.

$MaxExtraPrecision = 300; m = 300; f[p_] := 1 - 1/(2*p^2); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n]), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

PARI
```
prodeulerrat(1 - 1/(2*p^2))
```

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A056671(k)/A034444(k).

Equals Product_{p prime} (1 - 1/(2*p^2)).

A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 10, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 30, 5, 26, 30, 14, 29, 30, 31, 34, 33, 34, 35, 6, 37, 38, 39, 50, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 60, 55, 70, 57, 58, 59, 30, 61, 62, 21, 10, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A384559 at n = 512: a(512) = 522, while A384559(512) = 514.
The number of these divisors is A368979(n), and the largest of them is A331737(n).
The indices of records of a(n)/n are the primorial numbers (A002110) cubed, i.e., 1 and the terms of A115964.

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

Cf. A002110, A005117, A051377, A072587, A082020, A115964, A268335, A331737, A368979, A374459, A384559.

A384558:=proc(n)
    local a, pe,p,e,af,d;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        af := 0 ;
        for d in numtheory[divisors](e) do
            if type(d,'odd') then
                af := af+p^d ;
            end if;
        end do:
        a := a*af ;
    end do;
    a
end proc:
seq(A384558(n), n=1..100); # R. J. Mathar, Jun 04 2025

f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, (d % 2) * f[i,1]^d));}

Multiplicative with a(p^e) = Sum_{d|e, d odd} p^d.
a(n) = n if and only if n is squarefree (A005117).
a(n) < n if and only if n is in A072587.
a(n) > n if and only if n is in A374459.
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p) * Sum_{k>=1} p^(2*k+1)/(p^(4*k+2)-1)) = 0.80824764393216997768... .

A059479 Number of 3 X 3 matrices with elements from {0,...,n-1} such that the middle element of each of the eight lines of three (rows, columns and diagonals) is the square (mod n) of the difference of the end elements.

Original entry on oeis.org

1, 4, 9, 64, 25, 36, 49, 256, 729, 100, 121, 576, 169, 196, 225, 4096, 289, 2916, 361, 1600, 441, 484, 529, 2304, 15625, 676, 6561, 3136, 841, 900, 961, 16384, 1089, 1156, 1225, 46656, 1369, 1444, 1521, 6400, 1681, 1764, 1849, 7744, 18225, 2116, 2209

Offset: 1

John W. Layman, Feb 15 2001

This sequence is multiplicative. - Mitch Harris, Apr 19 2005

The sequence enumerates the solutions of a system of polynomials equations modulo n, hence is multiplicative by the Chinese Remainder Theorem. The middle entry of the 3 X 3 is zero modulo n. - Michael Somos, Apr 30 2005

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

Cf. A008833, A013664, A082020.

f[p_, e_] := p^(3*e - (Mod[e, 2])); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=if(n<1,0,n^3/core(n)) /* Michael Somos, Apr 30 2005 */

a(n) = A008833(n)*n^2, where A008833(n) is the largest square that divides n.
Multiplicative with a(p^e) = p^(3e - (e % 2)). - Mitch Harris, Jun 09 2005
Dirichlet g.f.: zeta(s-2)*zeta(2s-6)/zeta(2s-4). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ zeta(3/2) * n^(7/2) / (7*zeta(3)). - Vaclav Kotesovec, Sep 16 2020
Sum_{n>=1} 1/a(n) = 15*zeta(6)/Pi^2 = A082020 * A013664 = 1.546176... . - Amiram Eldar, Nov 03 2022

A380857 Squares of numbers that are neither squarefree nor prime powers.

Original entry on oeis.org

144, 324, 400, 576, 784, 1296, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736

Offset: 1

Michael De Vlieger, Feb 06 2025

Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).

Does not intersect A362605.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).

Cf. A013661, A082020, A085548, A154945.

Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2

isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
apply(sqr, select(isok, [1..200])) \\ Michel Marcus, Feb 07 2025

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A380857(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+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    return bisection(f,n,n)**2 # Chai Wah Wu, Feb 08 2025

a(n) = A126706(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - Amiram Eldar, Feb 08 2025

A383647 Decimal expansion of 15/(2*Pi^4).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, May 03 2025

			0.07699486691013251391864587450339020606370851390...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 64.

Jakub Buczak, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A030059, A082020, A088245, A088246, A092425, A151927.

Join[{0},RealDigits[15/(2Pi^4),10,100][[1]]]

Equals Sum_{n > 0} 1/A030059(n)^4.

Equals 10/A151927. - Hugo Pfoertner, May 03 2025

A335204 Decimal expansion of Product_{p = prime} (1 + 6/p^2).

Original entry on oeis.org

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

Offset: 1

Jude Thaddeus Poole Jr., Andrew Hinton, Reid Huntley, May 26 2020

			6.95743589392521746246609006782918530413...

Pieter Moree, Approximation of singular series and automata, manuscripta mathematica, Vol. 101 (2000), pp. 385-399.
Wikipedia, Euler Product.

Cf. A000040, A001248, A082020, A065474, A206256, A328017.

$MaxExtraPrecision = 10000; Do[Print[5/2*Exp[-N[Sum[(-1)^j*6^j*(PrimeZetaP[2*j] - 1/4^j)/j, {j, 1, t}], 120]]], {t, 100, 1000, 100}] (* Vaclav Kotesovec, May 29 2020 *)

prodeulerrat(1 + 6/p^2) \\ Amiram Eldar, Mar 17 2021

More terms from Vaclav Kotesovec, May 29 2020

cp's OEIS Frontend

A383057 Decimal expansion of the asymptotic mean of A056671(k)/A034444(k), the ratio between the number of squarefree unitary divisors and the number of unitary divisors over the positive integers.

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A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

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A059479 Number of 3 X 3 matrices with elements from {0,...,n-1} such that the middle element of each of the eight lines of three (rows, columns and diagonals) is the square (mod n) of the difference of the end elements.

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A380857 Squares of numbers that are neither squarefree nor prime powers.

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A383647 Decimal expansion of 15/(2*Pi^4).

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A335204 Decimal expansion of Product_{p = prime} (1 + 6/p^2).

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