A096624 -id:A096624 - cp's OEIS frontend

A052434 Nearest integer to R(n) - pi(n), where R(x) is the Riemann prime counting function.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0

Offset: 2

Eric W. Weisstein

sign

The Riemann prime counting function R(n) = Sum_{prime powers p^k <= n} 1/k = A096624(n)/A096625(n). - N. J. A. Sloane, Feb 07 2023

			a(13) = 0 because R(13) = 5.504 and pi(13) = 6.

Harry J. Smith, Table of n, a(n) for n = 2..10000
H. J. Smith, XPCalc - Extra Precision Floating-Point Calculator [Broken link]
Eric Weisstein's World of Mathematics, Riemann Prime Counting Function

Cf. A052435, A096624, A096625.

a=Round(Ri(n)-Pi(n)) - Harry J. Smith, Dec 31 2008

Corrected 6 terms, a(2), a(7), a(10), a(13), a(20) and a(48). Each was made 1 larger. Also gave an example for a(13) and a program for computing a(n). - Harry J. Smith, Dec 31 2008

A096625 Denominators of the Riemann prime counting function.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 3, 3, 3, 3, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60

Offset: 1

Eric W. Weisstein, Jul 01 2004

			0, 1, 2, 5/2, 7/2, 7/2, 9/2, 29/6, 16/3, 16/3, 19/3, ...

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 167.

Eric Weisstein's World of Mathematics, Riemann Prime Counting Function

Cf. A096624.

Table[Sum[PrimePi[x^(1/k)]/k, {k, Log2[x]}], {x, 100}] // Denominator (* Eric W. Weisstein, Jan 09 2019 *)

a(n) = denominator(sum(k=1, n, if (p=isprimepower(k), 1/p))); \\ Michel Marcus, Jan 07 2019

a(n) = denominator(sum(k=1, logint(n, 2), primepi(sqrtnint(n, k))/k)); \\ Daniel Suteu, Jan 07 2019

A096623 Decimal expansion of Integral_{t>=2} 1/(t*log(t)(t^2-1)) dt.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 01 2004

Maximum value of the integral in the Riemann prime counting function.

			0.1400101011432869266869173052342997331775279281270658289489468743113049149...

John Derbyshire, Prime Obsession, Joseph Henry Press, 2003, pp. 328-329.
Bernhard Riemann, On the Number of Prime Numbers less than a Given Quantity, 1859.

Robert G. Wilson v, Table of n, a(n) for n = 0..2509
Eric Weisstein's World of Mathematics, Riemann Prime Counting Function

Cf. A096624, A096625.

evalf(Integrate(1/(x*log(x)*(x^2-1)), x = 2..infinity), 120); # Vaclav Kotesovec, Feb 13 2019

RealDigits[ NIntegrate[1/(t Log[t](t^2 - 1)), {t, 2, Infinity}, MaxRecursion -> 8, AccuracyGoal -> 115, WorkingPrecision -> 128]][[1]] (* Robert G. Wilson v, Jul 05 2004 *)

default(realprecision, 120); intnum(x=2, oo, 1/(x*log(x)*(x^2 - 1))) \\ Vaclav Kotesovec, Feb 13 2019

A322713 a(n) = numerator of the Riemann prime counting function for 10^n.

Original entry on oeis.org

0, 16, 428, 445273, 56175529, 991892879, 18296822833013, 3559637526370229, 6427431691337929, 14804074778750628149, 9387415960571046321167, 594663752918349842404169, 200936708396848319452718531, 296345083061712053722716462103, 30189234512048649753828116713823

Offset: 0

Daniel Suteu, Dec 24 2018

			0, 16/3, 428/15, 445273/2520, 56175529/45045, 991892879/102960, 18296822833013/232792560, ...

Daniel Suteu, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Riemann Prime Counting Function

The corresponding denominators are A322714.

Cf. A000720, A096624.

a(n) = numerator(sum(k=1, logint(10^n, 2), primepi(sqrtnint(10^n, k))/k));

a(n) = A096624(10^n).

a(n) = numerator of Sum_{k=1..floor(log_2(10^n))} pi(floor(10^(n/k)))/k, where pi(x) is the prime counting function A000720.

cp's OEIS Frontend

A052434 Nearest integer to R(n) - pi(n), where R(x) is the Riemann prime counting function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

XPCalc

Extensions

A096625 Denominators of the Riemann prime counting function.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A096623 Decimal expansion of Integral_{t>=2} 1/(t*log(t)(t^2-1)) dt.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A322713 a(n) = numerator of the Riemann prime counting function for 10^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula