Benoit Cloitre - cp's OEIS frontend

A372971 a(1)=1, then a(n) = floor(n/min(a(n-1),a(floor(n/2)))).

1, 2, 3, 2, 2, 3, 2, 4, 4, 5, 5, 4, 4, 7, 7, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 4, 7, 4, 7, 8, 8, 8, 8, 9, 9, 9, 9, 8, 8, 10, 10, 8, 9, 11, 11, 8, 8, 8, 8, 8, 8, 9, 9, 14, 14, 8, 8, 15, 15, 8, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 8, 9, 8, 10

Offset: 1

Benoit Cloitre, May 18 2024

It seems that limsup and liminf of a(n)/sqrt(n) exist (see link).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Benoit Cloitre, Plot of a(n)/sqrt(n) for n=1 up to 400000
Hugo Pfoertner, Plot of a(n)/sqrt(n), n=1..400000, zoom into pdf to see details.

Cf. A372970, A097051.

a[1]=1; a[n_]:=Floor[n/Min[a[n-1],a[Floor[n/2]]]]; Array[a,80] (* Stefano Spezia, May 18 2024 *)

a(n)=if(n<2,1,floor(n/min(a(n-1),a(n\2))))

A372970 a(1)=1, then a(n) = floor(n/max(a(n-1),a(floor(n/2)))).

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 2, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 6, 4, 6, 4, 7, 4, 7, 4, 8, 4, 8, 4, 9, 4, 9, 4, 10, 4, 10, 4, 8, 5, 9, 5, 8, 6, 8, 6, 8, 6, 9, 6, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 9, 8, 8, 8, 10, 8, 8, 8, 10, 8, 11, 8, 11, 8, 10, 9, 10, 9, 10, 9, 10, 9, 11

Offset: 1

Benoit Cloitre, May 18 2024

nonn

It seems that limsup and liminf of a(n)/sqrt(n) exist (see link).

Benoit Cloitre, Plot of a(n)/sqrt(n) for n=1 up to 400000.
Hugo Pfoertner, Plot of a(n)/sqrt(n), n=1..400000, zoom into pdf to see details.

Cf. A372971, A097051.

a(n)=if(n<2,1,floor(n/max(a(n-1),a(n\2))))

A369633 Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jan 28 2024

			0.18707307250977978945095915767776663195781480296221...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Problems in Mathematical Analysis, Springer, 2013. See p. 100.

Cf. A074962, A345208.

RealDigits[3*Log[2*Pi]/2 - 6*Log[Glaisher] - EulerGamma - 1/2, 10, 120][[1]] (* Amiram Eldar, Jan 28 2024 *)

3*log(2*Pi)/2 + 6*zeta'(-1) - Euler - 1 \\ Amiram Eldar, Jan 28 2024

Integral_{x=0..1} frac(1/x)^3 dx = (3/2)*log(2*Pi) - 6*log(A) - gamma - 1/2 = 0.1870730725..., where A is the Glaisher-Kinkelin constant.
Equals 3*log(2) - 3/2 + 3 * Sum_{k>=1} ((-1)^k/(k+3))*(zeta(k+1)-1).
From Vaclav Kotesovec, Jan 29 2024: (Start)
Equals 6 * Sum_{k>=1} (zeta(k+1) - 1) / ((k+1)*(k+2)*(k+3)).
Equals -1/2 + 6 * Sum_{k>=2} zeta(k) / (k*(k+1)*(k+2)). (End)

A356800 Numbers m for which Sum_{k=1..m} 1/k^s has no zero in the half-plane Re(s)>1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 28

Offset: 1

Benoit Cloitre, Aug 28 2022

David J. Platt and Timothy S. Trudgian, Zeroes of partial sums of the Zeta function, LMS Journal of Computation and Mathematics, Volume 19, Issue 1, 2016, pp. 37-41.

A356216 Decimal expansion of the real part of the first nontrivial zero of zeta'.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Aug 13 2022

The nontrivial zero of zeta' with the smallest imaginary part is 2.4631618694543212... + i*23.2983204927628579...

The Riemann Hypothesis is equivalent to the assertion that zeta' (the derivative of the Riemann zeta function) has no nontrivial zero in the half-plane Re(z) < 1/2 (there are trivial zeros, e.g., -2.717262829204574...).

			2.463161869454321285874395053306329144920793134567323475022217370727117508671...

Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Mathematica, Vol. 133 (1974), pp. 49-65; alternative link.

Cf. A356092.

RealDigits[Re[x /. FindRoot[Derivative[1][Zeta][x], {x, 2 + 23*I}, WorkingPrecision -> 100]]][[1]] (* Amiram Eldar, Aug 14 2022 *)

A356092 Decimal expansion of the imaginary part of the first nontrivial zero of zeta'.

Original entry on oeis.org

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

Offset: 2

Benoit Cloitre, Aug 13 2022

The nontrivial zero of zeta' with the smallest imaginary part is 2.4631618694543212... + i*23.2983204927628579...

The Riemann Hypothesis is equivalent to the assertion that zeta' has no nontrivial zero in the half-plane Re(z) < 1/2 (there are trivial zeros, e.g., -2.717262829204574...).

Benoit Cloitre, Picture of some nontrivial zeros of zeta'.
Norman Levinson and Hugh L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Mathematica, Vol. 133 (1974), pp. 49-65; alternative link.

Cf. A356216.

RealDigits[Im[x /. FindRoot[Derivative[1][Zeta][x], {x, 2 + 23*I}, WorkingPrecision -> 100]]][[1]] (* Amiram Eldar, Aug 14 2022 *)

A356005 Number of integers k such that k*tau(k) <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 21, 22, 22, 22, 22, 22

Offset: 1

Benoit Cloitre, Jul 22 2022

nonn

Partial sums of A327166. - Rémy Sigrist, Jul 23 2022

R. Balasubramanian and K. Ramachandra, On the number of integers n such that nd(n)<=x, Acta Arith., 49 (1988), 313-322.
Vaclav Kotesovec, Plot of a(n) / (n/sqrt(log(n))) for n = 1..10^7

Cf. A038040, A327166.

Table[Sum[If[k*DivisorSigma[0, k] <= n, 1, 0], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jul 23 2022 *)

PARI
```
a(n)=sum(k=1,n,if(k*numdiv(k)<=n,1,0))
```

a(n) is asymptotic to C*n/sqrt(log(n)) for a suitable constant C > 0.

A350459 Number of positive rational points on the unit circle with denominator <= n and numerator >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 4, 4, 5, 5, 10, 10, 11, 11, 11, 19, 19, 19, 19, 19, 29, 32, 32, 32, 33, 44, 44, 44, 44, 47, 60, 60, 61, 61, 66, 82, 83, 83, 83, 83, 100, 100, 100, 100, 100, 122, 127, 134, 135, 135, 156, 156, 156, 159, 159, 183, 184, 184, 184, 184, 220, 220, 220, 227, 227, 254

Offset: 1

Benoit Cloitre, Jan 01 2022

nonn

A rational point (x,y) is of the form (a/c, b/d) with (a,b,c,d) integers. Sequence gives the number of quadruples (a,b,c,d) satisfying a >= b >= 1, 1 <= c <= n, 1 <= d <= n and such that a^2/c^2 + b^2/d^2 = 1.

Cf. A046109.

a(n)=sum(a=1,n,sum(b=1,a,sum(c=1,n,sum(d=1,n,if(a^2/c^2+b^2/d^2-1,0,1)))))

def A350459(n): return sum(1 for d in range(1,n+1) for c in range(1,n+1) for b in range(1,d+1) for a in range(1, b+1) if (a*d)**2 + (b*c)**2 == (c*d)**2) # Chai Wah Wu, Jan 19 2022

A349229 a(n) = Sum_{k=1..n} (-1)^A001222(k)*(-1)^A001222(k+1).

Original entry on oeis.org

-1, 0, -1, -2, -3, -4, -3, -4, -3, -4, -3, -2, -3, -2, -1, -2, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 6, 7, 8, 9, 10, 9, 8, 9, 8, 7, 6, 5, 6, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 5, 4, 5, 6, 7, 6, 7, 8, 7, 6

Offset: 1

Benoit Cloitre, Nov 11 2021

sign

It is conjectured that a(n)=o(n).

Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, An averaged form of Chowla's conjecture, Algebra & Number Theory, Vol. 9, No. 9 (2015), pp. 2167-2196.

Cf. A001222.

a[n_] := Sum[(-1)^(PrimeOmega[k] + PrimeOmega[k + 1]), {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Nov 11 2021 *)

a(n)=sum(k=1,n,(-1)^bigomega(k)*(-1)^bigomega(k+1))

A347727 a(1)=2; then a(n) is the least integer > a(n-1) such that 2 is the largest element in the continued fraction for 1/a(1) + 1/a(2) + ... + 1/a(n).

Original entry on oeis.org

2, 6, 18, 102, 40936, 4252528, 7112715120

Offset: 1

Benoit Cloitre, Sep 11 2021

3.5*10^15 < a(8) <= 8778368652367133280. - Jon E. Schoenfield, Sep 12 2021

			contfrac(1/2 + 1/6 + 1/18 + 1/102 + 1/40936) = [0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2] and 1/2 + 1/6 + 1/18 + 1/102 + 1/40936 = sqrt(3) - 1.0000002354...

Cf. A160390.

a[1] = 2; a[n_] := a[n] = Module[{k = a[n - 1] + 1, s = Sum[1/a[k], {k, 1, n - 1}]}, While[Max[ContinuedFraction[s + 1/k]] != 2, k++]; k]; Array[a, 6] (* Amiram Eldar, Sep 11 2021 *)

a(7) from Jon E. Schoenfield, Sep 11 2021

cp's OEIS Frontend

User: Benoit Cloitre

A372971 a(1)=1, then a(n) = floor(n/min(a(n-1),a(floor(n/2)))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A372970 a(1)=1, then a(n) = floor(n/max(a(n-1),a(floor(n/2)))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369633 Decimal expansion of integral of frac(1/x)^3 dx for x=0 to 1.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A356800 Numbers m for which Sum_{k=1..m} 1/k^s has no zero in the half-plane Re(s)>1.

Views

Author

Keywords

Links

A356216 Decimal expansion of the real part of the first nontrivial zero of zeta'.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A356092 Decimal expansion of the imaginary part of the first nontrivial zero of zeta'.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A356005 Number of integers k such that k*tau(k) <= n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A350459 Number of positive rational points on the unit circle with denominator <= n and numerator >= 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Python

A349229 a(n) = Sum_{k=1..n} (-1)^A001222(k)*(-1)^A001222(k+1).

Views

Author

Keywords