A153810 -id:A153810 - cp's OEIS frontend

A242493 a(n) is the number of not-sqrt-smooth numbers ("jagged" numbers) not exceeding n. This is the counting function of A064052.

0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 15, 16, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 32, 32, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 41, 42, 43, 44, 45

Offset: 1

Jean-François Alcover, May 16 2014

nonn

This sequence is different from shifted A072490, after 22 terms.

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
Daniel H. Greene and Donald E. Knuth, Mathematics for the Analysis of Algorithms, 3rd ed., Birkhäuser, 1990, pp. 95-98.

Cf. A064052, A064775, A072490, A153810, A242610.

jaggedQ[n_] := jaggedQ[n] = (f = FactorInteger[n][[All, 1]]; s = Sqrt[n]; Count[f, p_ /; p > s] > 0); a[n_] := ( For[ cnt = 0; j = 2, j <= n, j++, If[jaggedQ[j], cnt++]]; cnt); Table[a[n], {n, 1, 100}]

from math import isqrt
from sympy import primepi
def A242493(n): return sum(primepi(n//i)-primepi(i) for i in range(1,isqrt(n)+1)) # Chai Wah Wu, Sep 01 2024

From Ridouane Oudra, Nov 07 2019: (Start)
a(n) = Sum_{i=1..floor(sqrt(n))} (pi(floor(n/i)) - pi(i)).
a(n) = Sum_{p<=sqrt(n)} (p-1) + Sum_{sqrt(n)a(n) = n - A064775(n). (End)
a(n) ~ log(2)*n - A153810 * n/log(n) - A242610 * n/log(n)^2 + O(n/log(n)^3) (Greene and Knuth, 1990). - Amiram Eldar, Apr 15 2021

A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.18384376406702461207534175664658126707821355787059...

Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A002117, A002162, A002391, A153810, A386733, A386736, A386737.

RealDigits[9*Log[3]/2 - 6*Log[2] - Zeta[3]/2, 10, 120][[1]]

PARI
```
9*log(3)/2 - 6*log(2) - zeta(3)/2
```

Equals 9*log(3)/2 - 6*log(2) - zeta(3)/2.

A109996 Primes p such that the arithmetic mean of the fractional parts of p/1, p/2, ..., p/p is larger than 1 - gamma = 0.422784...

Original entry on oeis.org

23, 47, 53, 59, 71, 83, 89, 107, 131, 139, 149, 167, 179, 191, 223, 227, 239, 251, 263, 269, 293, 311, 317, 347, 349, 359, 383, 389, 419, 431, 439, 449, 461, 467, 479, 491, 503, 509, 557, 569, 571, 587, 593, 599, 607, 619, 643, 647, 659, 683, 701, 719, 727

Offset: 1

Stefan Krämer, Sep 01 2005

nonn

S. R. Finch. Mathematical Constants. Cambridge University Press, 2003 ISBN 0-521-81802-2 p. 29.
Stefan Kraemer. Eulers constant and related numbers, preprint, 2005.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Stefan Kraemer, Euler's Constant 0.577... Its Mathematics and History

Cf. A104885, A109997.

Cf. A153810 (1-gamma).

H:= proc(n) H(n):= 1/n+`if`(n=1, 0, H(n-1)) end:
a:= proc(n) option remember; local c, p; Digits := 1000;
      c:= evalf(1-gamma);
      p:=`if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if H(p)-add(iquo(p, i), i=1..p)/p>c
         then return p fi
      od
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jun 14 2013

Reap[For[p = 2, p < 1000, p = NextPrime[p], If[Mean[FractionalPart /@ (p/Range[p])] > 1-EulerGamma, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Dec 28 2021 *)

lista(nn) = {forprime(p=2, nn, if (sum (i=1, p, p/i - floor(p/i))/p > 1- Euler, print1(p, ", ")););} \\ Michel Marcus, Jun 14 2013

A242610 Decimal expansion of 1-gamma-gamma(1), a constant related to the asymptotic expansion of j(n), the counting function of "jagged" numbers, where gamma is Euler-Mascheroni constant and gamma(1) the first Stieltjes constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 19 2014

			0.495600180582143864254074285792498888...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 2.17, p. 102.

Ovidiu Furdui, Problem 164, Missouri J. Math. Sci., Vol. 18, No. 2 (2006), p. 148; Solution, ibid., Vol. 19, No. 2 (2007), pp. 156-158.

Cf. A064052, A082633, A153810, A242493.

RealDigits[1 - EulerGamma - StieltjesGamma[1], 10, 100] // First

j(n) = log(2)*n - (1-gamma)*n/log(n) - (1-gamma-gamma(1))*n/log(n)^2 + O(n/log(n)^3).
Equals -Integral_{x=0..1} frac(1/x)*log(x) dx (Furdui, 2007 and 2013). - Amiram Eldar, Mar 26 2022
Equals Integral_{x=0..1} Integral_{y=0..1} frac(1/(x*y)) dx dy (Furdui, 2013, section 2.43, p. 106). - Amiram Eldar, Jul 31 2025

A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.56382732769577740059825665959334054154152531811711...

Ovidui Furdui, Problem 150, Problems, Missouri J. Math. Sci., Vol. 16, No. 2 (2004), p. 130; Huizeng Qin, Solution to Problem 150, ibid., Vol. 17, No. 3 (2005), pp. 197-199.
Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A016627, A072691, A153810, A345208, A386734, A386735.

RealDigits[2*Log[2] - Pi^2/12, 10, 120][[1]]

PARI
```
2*log(2) - zeta(2)/2
```

Equals 2*log(2) - Pi^2/12 = A016627 - A072691.

A346367 Decimal expansion of (1 - gamma)*zeta(2), where gamma is Euler's constant (or the Euler-Mascheroni constant).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jul 14 2021

			0.695452355733244911926851064414320...

Cf. A001620, A013661, A153810, A346111, A346366.

First[RealDigits[(1-EulerGamma)Zeta[2],10,105]]

(1-Euler)*zeta(2) \\ Michel Marcus, Jul 16 2021

Equals A153810*A013661.

A386711 Decimal expansion of Sum_{k>=2} (zeta(k)-1)/(k+1).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 31 2025

			0.29245363434456082791641421855318114461752285839225...

Hari M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012. See eq. (500), p. 314.

Ovidiu Furdui, The evaluation of a class of fractional part integrals, Integral Transforms and Special Functions, Vol. 26, No. 8 (2015), pp. 635-641.
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 348.
Hari M. Srivastava and Junesang Choi, Series Associated with the Zeta and Related Functions, Springer Science+Business Media Dordrecht, 2001. See eq. (474), p. 213.

Cf. A001620 (gamma), A061444.

Sum_{k>=2} (zeta(k)-1)/(k+m): A153810 (m=0), this constant (m=1), A386712 (m=2).

RealDigits[3/2 - EulerGamma/2 - Log[2*Pi]/2, 10, 120][[1]]

PARI
```
3/2 - Euler/2 - log(2*Pi)/2
```

Equals 3/2 - gamma/2 - log(2*Pi)/2 (Srivastava and Choi, 2001).

Equals -Sum_{k>=2} (k*log(1-1/k) + 1 + 1/(2*k)) (Shamos, 2011).

A386712 Decimal expansion of Sum_{k>=2} (zeta(k)-1)/(k+2).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 31 2025

			0.22448062442724777958966024641468693092980998704517...

Hari M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012. See eq. (543), p. 320.

Ovidiu Furdui, The evaluation of a class of fractional part integrals, Integral Transforms and Special Functions, Vol. 26, No. 8 (2015), pp. 635-641.
Michael I. Shamos, Shamos's catalog of the real numbers, 2011. See p. 290.
Hari M. Srivastava and Junesang Choi, Series Associated with the Zeta and Related Functions, Springer Science+Business Media Dordrecht, 2001. See eq. (517), p. 219.

Cf. A001620 (gamma), A061444, A074962 (A), A225746.

Sum_{k>=2} (zeta(k)-1)/(k+m): A153810 (m=0), A386711 (m=1), this constant (m=2).

RealDigits[11/6 - EulerGamma/3 - 2*Log[Glaisher] - Log[2*Pi]/2, 10, 120][[1]]

11/6 - Euler/3 - 2*(1/12-zeta'(-1)) - log(2*Pi)/2

Equals 11/6 - gamma/3 - 2*log(A) - log(2*Pi)/2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant (Srivastava and Choi, 2001).

Equals -Sum_{k>=2} (k^2*log(1-1/k) + k + 1/(3*k) + 1/2) (Shamos, 2011).

A319026 Decimal expansion of Psi(3).

Original entry on oeis.org

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

Offset: 0

Patrick C. Schneider, Sep 08 2018

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			0.92278433509846713939348790...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, pp. 258-259 [alternative scanned copy].

Cf. A001620, A153810.

evalf(Psi(3.0)) ; - R. J. Mathar, Aug 29 2023

RealDigits[3/2 - EulerGamma, 10, 100][[1]] (* Amiram Eldar, May 24 2023 *)

PARI
```
psi(3)
```

Psi(3) = Psi(2) + 1/2 = 3/2 - gamma, with gamma = A001620 and Psi(2) = A153810. - Wolfdieter Lang, Oct 05 2018

Showing 1-10 of 12 results. Next

The first two formulae (in the Formula section) are similar to the sum and integral lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k) = Integral_{x=0..1} frac(1/x) dx = 1 - gamma (A153810).

The second raw moment of the distribution of the fractional part of 1/x, where x is chosen uniformly at random from (0, 1]. Since the expected value is 1 - gamma, the second central moment, or variance, is log(2*Pi) - gamma - 1 - (1 - gamma)^2 = log(2*Pi) - gamma^2 + gamma - 2 = 0.081914807503... and the standard deviation is sqrt(log(2*Pi) - gamma^2 + gamma - 2) = 0.2862076300...

cp's OEIS Frontend

A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).

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A242493 a(n) is the number of not-sqrt-smooth numbers ("jagged" numbers) not exceeding n. This is the counting function of A064052.

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A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

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A109996 Primes p such that the arithmetic mean of the fractional parts of p/1, p/2, ..., p/p is larger than 1 - gamma = 0.422784...

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A242610 Decimal expansion of 1-gamma-gamma(1), a constant related to the asymptotic expansion of j(n), the counting function of "jagged" numbers, where gamma is Euler-Mascheroni constant and gamma(1) the first Stieltjes constant.

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A386733 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} {1/(x+y)} dx dy, where {} denotes fractional part.

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A346367 Decimal expansion of (1 - gamma)*zeta(2), where gamma is Euler's constant (or the Euler-Mascheroni constant).

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A386711 Decimal expansion of Sum_{k>=2} (zeta(k)-1)/(k+1).

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