A096625 -id:A096625 - 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

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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

A096624 Numerators of the Riemann prime counting function.

Original entry on oeis.org

0, 1, 2, 5, 7, 7, 9, 29, 16, 16, 19, 19, 22, 22, 22, 91, 103, 103, 115, 115, 115, 115, 127, 127, 133, 133, 137, 137, 149, 149, 161, 817, 817, 817, 817, 817, 877, 877, 877, 877, 937, 937, 997, 997, 997, 997, 1057, 1057, 1087, 1087, 1087, 1087, 1147, 1147, 1147

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. A096625.

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

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

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

Let Sk{f(k)}= Sum_{k>=2}f(k), then the g.f. of A096624/A096625 can be written as
(1/1)*Sa{(x^a)/(1-x)} - (1/2)*Sa{ Sb{ (x^(a*b))/(1-x)}} + (1/3)*Sa{ Sb{ Sc{ (x^(a*b*c))/(1-x)}}} - (1/4)*Sa{ Sb{ Sc{ Sd{ (x^(a*b*c*d))/(1-x)}}}} + ... . - Mats Granvik, Apr 06 2011
a(n)/A096625(n) = Sum_{p prime <= n} HarmonicNumber(floor(log_p n)). - Ammar Khatab, Jan 25 2025

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

A322714 a(n) = denominator of the Riemann prime counting function for 10^n.

Original entry on oeis.org

1, 3, 15, 2520, 45045, 102960, 232792560, 5354228880, 1115464350, 291136195350, 20629078984800, 144403552893600, 5342931457063200, 856326196254765600, 9419588158802421600, 3099044504245996706400, 4106233968125945635980, 16424935872503782543920

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 numerators are A322713.

Cf. A000720, A096625.

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

a(n) = A096625(10^n).

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

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

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A096624 Numerators of the Riemann prime counting function.

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

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A322714 a(n) = denominator of the Riemann prime counting function for 10^n.

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