A152416 -id:A152416 - cp's OEIS frontend

A243351 Difference between 2n and the n-th squarefree number: a(n) = 2n - A005117(n).

1, 2, 3, 3, 4, 5, 4, 5, 5, 6, 7, 7, 7, 7, 8, 9, 8, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 14, 15, 13, 13, 13, 13, 14, 15, 15, 16, 15, 16, 17, 17, 18, 19, 19, 20, 19, 20, 21, 20, 21, 21, 22, 23, 23, 23, 23, 24, 25, 25, 23, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30

Offset: 1

Antti Karttunen, Jun 04 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A005117, A152416, A243348.

Module[{sf=Select[Range[150],SquareFreeQ]},Table[2n-sf[[n]],{n,Length[ sf]}]] (* Harvey P. Dale, Jun 26 2021 *)

from math import isqrt
from sympy import mobius
def A243351(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return (n<<1)-m # Chai Wah Wu, Aug 12 2024

a(n) = 2n - A005117(n).
a(n) = n - A243348(n).
a(n) ~ c * n, where c = 2 - Pi^2/6 (A152416). - Amiram Eldar, Mar 04 2024

A354238 Decimal expansion of 1 - Pi^2/12.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, May 20 2022

Ratio of area between the polygon that is adjacent in the same plane to the base of the stepped pyramid with an infinite number of levels described in A245092 and the circumscribed square (see the first formula).

			0.177532966575886781763792416676987405390525049396600781132220885314996264798...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013.

Ovidiu Furdui, Problem A, Nieuw Archief voor Wiskunde, Vol. 9, No. 1 (2008), p. 86; "Problem 2008/1-A, Solution to Problem A by Noud Aldenhoven and Daan Wanrooy, ibid., Vol. 9, No. 3 (2008), p. 303.
Ovidiu Furdui, Problem 1930, Mathematics Magazine, Vol. 86, No. 4 (2013), p. 289; A zeta series, Solution to Problem 1930 by Omran Kouba, ibid., Vol. 87, No. 4 (2014), pp. 296-298.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C7.4).
Index entries for transcendental numbers.

Cf. A000290, A004125, A013661, A024916, A072691, A152416, A237593, A245092, A353908.

RealDigits[1 - Pi^2/12, 10, 100][[1]] (* Amiram Eldar, May 20 2022 *)

PARI
```
1-Pi^2/12
```
PARI
```
1-zeta(2)/2
```

Equals lim_{n->infinity} A004125(n)/(n^2).
Equals 1 - A013661/2.
Equals 1 - A072691.
Equals A152416/2.
Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - Amiram Eldar, May 20 2022
Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013 problem). - Amiram Eldar, Jun 09 2022
Equals Integral_{x>=1} {x}/x^3 dx where {.} is the fractional part. [Nahin]. R. J. Mathar, May 22 2024
From Amiram Eldar, Jul 31 2025: (Start)
Equals Integral_{x=0..1} {1/x} * x dx (Furdui, 2013 book, section 2.21, page 103).
Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}*{y/x} dx dy, where {} denotes fractional part (Furdui, 2008 and 2013 book, section 2.36, page 105). (End)

A152419 Decimal expansion of 3-Pi^2/6-zeta(3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 03 2008

Consider the constants N(s) = Sum_{n>=2} 1/(n^s*(n-1)) = s-Sum_{k=2..s} zeta(k), where zeta() is Riemann's zeta function. We have N(1)=1 and this constant here is N(3).

			0.15300902999217927812784667184252482001606380645270268047217021528815...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009, section 4.1.

Cf. A013661 (Pi^2/6), A002117 (zeta(3)).

Cf. A152416.

Maple
```
evalf(3-Pi^2/6-Zeta(3));
```

RealDigits[3-Pi^2/6-Zeta[3],10,120][[1]] (* Harvey P. Dale, Jul 01 2022 *)

3-Pi^2/6-zeta(3) \\ Charles R Greathouse IV, Jan 31 2017

t(n) = 1/(n*(n+1)^(3));
sum(t(n), n, 1, oo).n(digits=107); # Jani Melik, Nov 20 2020

Equals 3-A013661-A002117.

A262605 Decimal expansion of Integral_{0..1} log(1-x)*log(x)^2 dx (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 26 2015

			-0.30601805998435855625569334368504964003212761290540536094434 ...

M. Jung, Y. J. Cho, J. Choi, Euler sums evaluatable from integrals, Commun. Korean Math. Soc. 19 (2008), 545-555.

Cf. A152416 (Integral_{0..1} log(1-x)*log(x) dx), A262606 (Integral_{0..1} log(1-x)^2*log(x)^2 dx).

RealDigits[Integrate[Log[1 - x]*Log[x]^2, {x, 0, 1}] , 10,
  105] // First

-(-6 + Pi^2/3 + 2*zeta(3)) \\ Michel Marcus, Sep 27 2015

Equals -6 + Pi^2/3 + 2 zeta(3).

Equals Integral_{0..Pi/2} log(cos(x)^2) * log(sin(x)^2)^2 * sin(2x) dx.

A262606 Decimal expansion of Integral_{0..1} log(1-x)^2*log(x)^2 dx (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 26 2015

			0.141749006226296033506769678199030657353759499702894536 ...

M. Jung, Y. J. Cho, J. Choi, Euler sums evaluatable from integrals, Commun. Korean Math. Soc. 19 (2008), 545-555.

Cf. A152416 (Integral_{0..1} log(1-x)*log(x) dx), A262605 (Integral_{0..1} log(1-x)*log(x)^2 dx).

RealDigits[24 - 4*Pi^2/3 - Pi^4/90 - 8 Zeta[3], 10, 105] // First

24 - 4*Pi^2/3 - Pi^4/90 - 8*zeta(3) \\ Michel Marcus, Sep 27 2015

Equals 24 - 4 Pi^2/3 - Pi^4/90 - 8 zeta(3).

Also equals Integral_{0..Pi/2} log(cos(x)^2)^2 * log(sin(x)^2)^2 * sin(2x) dx.

A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jun 13 2024

			0.69010488251022497818773002567827532644066623131...

Steve Fan, The shifted prime-divisor function over shifted primes, arXiv:2406.05217 [math.NT], 2024. See p. 34.

Cf. A002117, A013661, A013664, A082695, A152416, A157289, A157292, A183699.

RealDigits[(2-Zeta[2])Zeta[2]Zeta[3]/Zeta[6],10,100][[1]]

cp's OEIS Frontend

A243351 Difference between 2n and the n-th squarefree number: a(n) = 2n - A005117(n).

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A354238 Decimal expansion of 1 - Pi^2/12.

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A152419 Decimal expansion of 3-Pi^2/6-zeta(3).

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A262605 Decimal expansion of Integral_{0..1} log(1-x)*log(x)^2 dx (negated).

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A262606 Decimal expansion of Integral_{0..1} log(1-x)^2*log(x)^2 dx (negated).

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A373702 Decimal expansion of (2 - zeta(2))*zeta(2)*zeta(3)/zeta(6).

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A373702 Decimal expansion of (2 - zeta(2))zeta(2)zeta(3)/zeta(6).