A257099 -id:A257099 - cp's OEIS frontend

A257098 From square root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose square is 1/zeta; sequence gives numerator of b(n).

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -5, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -7, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 5, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -21, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 5, -5, 1, -1, -1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 7, -1, 1, 1, 1

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = A257098(n)/A046644(n) is (zeta(x))^(-1/2).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/2).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...
The sequence of rationals a(n)/A046644(n) is the Moebius transform of A046643/A046644 which is multiplicative. This sequence is then also multiplicative. - Andrew Howroyd, Aug 08 2018

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)
Vaclav Kotesovec, Graph - the asymptotic ratio (10^7 terms)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 2;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1]==1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257098 *)
den = Denominator[t] (* A046644 *)

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
DirSqrt(v)={my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dAndrew Howroyd, Aug 08 2018

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 2;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257098(j)/A046644(j) ~ -n / (2 * sqrt(Pi) * log(n)^(3/2)) * (1 + 3*(gamma/2 + 1)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 05 2025

A257100 From fourth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, -7, -3, 1, -1, 3, -1, 1, 1, -77, -1, 3, -1, 3, 1, 1, -1, 7, -3, 1, -7, 3, -1, -1, -1, -231, 1, 1, 1, 9, -1, 1, 1, 7, -1, -1, -1, 3, 3, 1, -1, 77, -3, 3, 1, 3, -1, 7, 1, 7, 1, 1, -1, -3, -1, 1, 3, -1463, 1, -1, -1, 3, 1, -1, -1, 21, -1, 1, 3, 3, 1, -1, -1, 77, -77, 1, -1, -3, 1, 1, 1, 7, -1, -3, 1, 3, 1, 1, 1, 231, -1, 3, 3, 9

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = a(n)/A256691(n) is (zeta(x))^(-1/4).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/4).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 4;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257100 *)
den = Denominator[t] (* A256691 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/4))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 4;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257100(j)/A256691(j) ~ n / (Gamma(-1/4) * log(n)^(5/4)) * (1 + 5*(gamma/4 + 1)/(4*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A257101 From fifth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fifth power is 1/zeta; sequence gives numerator of b(n).

Original entry on oeis.org

1, -1, -1, -2, -1, 1, -1, -6, -2, 1, -1, 2, -1, 1, 1, -21, -1, 2, -1, 2, 1, 1, -1, 6, -2, 1, -6, 2, -1, -1, -1, -399, 1, 1, 1, 4, -1, 1, 1, 6, -1, -1, -1, 2, 2, 1, -1, 21, -2, 2, 1, 2, -1, 6, 1, 6, 1, 1, -1, -2, -1, 1, 2, -1596, 1, -1, -1, 2, 1, -1, -1, 12, -1, 1, 2, 2, 1, -1, -1, 21, -21, 1, -1, -2, 1, 1, 1, 6, -1, -2, 1, 2, 1, 1, 1, 399, -1, 2, 2, 4

Offset: 1

Wolfgang Hintze, Apr 16 2015

Dirichlet g.f. of b(n) = A257101(n)/A256693(n) is (zeta(x))^(-1/5).
Denominator is the same as for Dirichlet g.f. (zeta(x))^(+1/5).
Formula holds for general Dirichlet g.f. zeta(x)^(-1/k) with k = 1, 2, ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..500 from Wolfgang Hintze)
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. family zeta^(-1/k): A257098/A046644 (k=2), A257099/A256689 (k=3), A257100/A256691 (k=4), A257101/A256693 (k=5).

Cf. family zeta^(+1/k): A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

k = 5;
c[1, n_] = b[n];
c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
nn = 100; eqs = Table[c[k, n]==MoebiusMu[n], {n, 1, nn}];
sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
t = Table[b[n], {n, 1, nn}] /. sol[[1]];
num = Numerator[t] (* A257101 *)
den = Denominator[t] (* A256693 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(-1/5))[n]), ", ")) \\ Vaclav Kotesovec, May 04 2025

with k = 5;
zeta(x)^(-1/k) = Sum_{n>=1} b(n)/n^x;
c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
Then solve c(k,n) = mu(n) for b(m);
a(n) = numerator(b(n)).
Sum_{j=1..n} A257101(j)/A256693(j) ~ n / (Gamma(-1/5) * log(n)^(6/5)) * (1 + 6*(gamma/5 + 1)/(5*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - Vaclav Kotesovec, May 05 2025

A383793 Numerators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

Original entry on oeis.org

1, 2, 1, 8, 5, 2, 7, 112, 2, 10, 11, 8, 13, 14, 5, 560, 17, 4, 19, 40, 7, 22, 23, 112, 50, 26, 14, 56, 29, 10, 31, 2912, 11, 34, 35, 16, 37, 38, 13, 560, 41, 14, 43, 88, 10, 46, 47, 560, 98, 100, 17, 104, 53, 28, 55, 784, 19, 58, 59, 40, 61, 62, 14, 46592, 65

Offset: 1

Vaclav Kotesovec, May 10 2025

General formula: if k >= 0, m > 0, and the Dirichlet generating function is zeta(s-k)^m * f(s), where f(s) has all possible poles at points less than k+1, then Sum_{j=1..n} a(j) ~ n^(k+1) * log(n)^(m-1) * f(k+1) / ((k+1) * Gamma(m)) * (1 + (m-1)*(m*gamma - 1/(k+1) + f'(k+1)/f(k+1)) / log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A256688, A256689, A257099, A383705, A383794 (denominators).

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/3))[n]), ", "))

Sum_{k=1..n} A383793(k) / A383794(k) ~ n^2 / (2*Gamma(1/3)*log(n)^(2/3)) * (1 + (1 - 2*gamma/3)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the Gamma function.

A383794 Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

Original entry on oeis.org

1, 3, 1, 9, 3, 3, 3, 81, 1, 9, 3, 9, 3, 9, 3, 243, 3, 3, 3, 27, 3, 9, 3, 81, 9, 9, 3, 27, 3, 9, 3, 729, 3, 9, 9, 9, 3, 9, 3, 243, 3, 9, 3, 27, 3, 9, 3, 243, 9, 27, 3, 27, 3, 9, 9, 243, 3, 9, 3, 27, 3, 9, 3, 6561, 9, 9, 3, 27, 3, 27, 3, 81, 3, 9, 9, 27, 9, 9, 3, 729

Offset: 1

Vaclav Kotesovec, May 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A256688, A256689, A257099, A383705, A383793 (numerators).

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p*X)^(1/3))[n]), ", "))

cp's OEIS Frontend

A257098 From square root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose square is 1/zeta; sequence gives numerator of b(n).

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A257100 From fourth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is 1/zeta; sequence gives numerator of b(n).

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A257101 From fifth root of the inverse of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fifth power is 1/zeta; sequence gives numerator of b(n).

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A383793 Numerators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

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A383794 Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-1)^(1/3).

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