A013661 -id:A013661 - cp's OEIS frontend

A019670 Decimal expansion of Pi/3.

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

Offset: 1

N. J. A. Sloane, Dec 11 1996

With an offset of zero, also the decimal expansion of Pi/30 ~ 0.104719... which is the average arithmetic area  of the 0-winding sectors enclosed by a closed Brownian planar path, of a given length t, according to Desbois, p. 1. - Jonathan Vos Post, Jan 23 2011
Polar angle (or apex angle) of the cone that subtends exactly one quarter of the full solid angle. See comments in A238238. - Stanislav Sykora, Jun 07 2014
60 degrees in radians. - M. F. Hasler, Jul 08 2016
Volume of a quarter sphere of radius 1. - Omar E. Pol, Aug 17 2019
Also smallest positive zero of Sum_{k>=1} cos(k*x)/k = -log(2*|sin(x/2)|). Proof of this identity: Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i = sqrt(-1). - Jianing Song, Nov 09 2019
The area of a circle circumscribing a unit-area regular dodecagon. - Amiram Eldar, Nov 05 2020

			Pi/3 = 1.04719755119659774615421446109316762806572313312503527365831486...
From _Peter Bala_, Nov 16 2016: (Start)
Case n = 1. Pi/3 = 18 * Sum_{k >= 0} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ).
Using the methods of Borwein et al. we can find the following asymptotic expansion for the tails of this series: for N divisible by 6 there holds Sum_{k >= N/6} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ) ~ 1/N^3 + 6/N^5 + 1671/N ^7 - 241604/N^9 + ..., where the sequence [1, 0, 6, 0, 1671, 0, -241604, 0, ...] is the sequence of coefficients in the expansion of ((1/18)*cosh(2*x)/cosh(3*x)) * sinh(3*x)^2 = x^2/2! + 6*x^4/4! + 1671*x^6/6! - 241604*x^8/8! + .... Cf. A024235, A278080 and A278195. (End)

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.3, p. 489.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.
J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Jean Desbois and Stéphane Ouvry, Algebraic and arithmetic area for m planar Brownian paths, arXiv:1101.4135 [math-ph], Jan 21, 2011.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A019669 (Pi/2), A019692 (2*Pi), A122952 (3*Pi), A003881(Pi/4), A137914, A238238, A002388, A091925, A093954, A016910, A019694, A136017, A352324.

Integral_{x=0..oo} 1/(1+x^m) dx: A013661 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), this sequence (m=6), A352125 (m=8), A094888 (m=10).

RealDigits[N[Pi/3,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/3 \\ Charles R Greathouse IV, Sep 08 2011

N=150; default(realprecision, 3+N); digits(Pi*10^N\3) \\ M. F. Hasler, Jul 08 2016

A third of A000796, a sixth of A019692, the square root of A100044.
Sum_{k >= 0} (-1)^k/(6k+1) + (-1)^k/(6k+5). - Charles R Greathouse IV, Sep 08 2011
Product_{k >= 1}(1-(6k)^(-2))^(-1). - Fred Daniel Kline, May 30 2013
From Peter Bala, Feb 05 2015: (Start)
Pi/3 = Sum {k >= 0} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k = 2F1(1/2,1/2;3/2;1/4). Similar series expansions hold for Pi^2 (A002388), Pi^3 (A091925) and Pi/(2*sqrt(2)) (A093954.)
The integer sequences A(n) := 4^n*(2*n + 1)! and B(n) := A(n)*( Sum {k = 0..n} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k ) both satisfy the second-order recurrence equation u(n) = (20*n^2 + 4*n + 1)*u(n-1) - 8*(n - 1)*(2*n - 1)^3*u(n-2). From this observation we can obtain the continued fraction expansion Pi/3 = 1 + 1/(24 - 8*3^3/(89 - 8*2*5^3/(193 - 8*3*7^3/(337 - ... - 8*(n - 1)*(2*n - 1)^3/((20*n^2 + 4*n + 1) - ... ))))). Cf. A002388 and A093954. (End)
Equals Sum_{k >= 1} arctan(sqrt(3)*L(2k)/L(4k)) where L=A000032. See also A005248 and A056854. - Michel Marcus, Mar 29 2016
Equals Product_{n >= 1} A016910(n) / A136017(n). - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=-oo..oo} sech(x)/3 dx. - Ilya Gutkovskiy, Jun 09 2016
From Peter Bala, Nov 16 2016: (Start)
Euler's series transformation applied to the series representation Pi/3 = Sum_{k >= 0} (-1)^k/(6*k + 1) + (-1)^k/(6*k + 5) given above by Greathouse produces the faster converging series Pi/3 = (1/2) * Sum_{n >= 0} 3^n*n!*( 1/(Product_{k = 0..n} (6*k + 1)) + 1/(Product_{k = 0..n} (6*k + 5)) ).
The series given above by Greathouse is the case n = 0 of the more general result Pi/3 = 9^n*(2*n)! * Sum_{k >= 0} (-1)^(k+n)*( 1/(Product_{j = -n..n} (6*k + 1 + 6*j)) + 1/(Product_{j = -n..n} (6*k + 5 + 6*j)) ) for n = 0,1,2,.... Cf. A003881. See the example section for notes on the case n = 1.(End)
Equals Product_{p>=5, p prime} p/sqrt(p^2-1). - Dimitris Valianatos, May 13 2017
Equals A019699/4 or A019693/2. - Omar E. Pol, Aug 17 2019
Equals Integral_{x >= 0} (sin(x)/x)^4 = 1/2 + Sum_{n >= 0} (sin(n)/n)^4, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} 1/(1 + x^6) dx. - Bernard Schott, Mar 12 2022
Pi/3 = -Sum_{n >= 1} i/(n*P(n, 1/sqrt(-3))*P(n-1, 1/sqrt(-3))), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximation Pi/3 = 1.04719755(06...) correct to 8 decimal places. - Peter Bala, Mar 16 2024
Equals Integral_{x >= 0} (2*x^2 + 1)/((x^2 + 1)*(4*x^2 + 1)) dx. - Peter Bala, Feb 12 2025

A082020 Decimal expansion of 15/Pi^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 09 2003

3/(2*Pi^2) (the same decimal expansion with an offset 0) is the probability that the greatest common divisor of two numbers selected at random is 2 (Christopher, 1956). - Amiram Eldar, May 23 2020
Sum of 1/n^2 over all squarefree n, see Penn link. - Charles R Greathouse IV, Jan 01 2022
Equals the asymptotic mean of the abundancy index of the cubefree numbers (A004709) (Jakimczuk and Lalín, 2022). - Amiram Eldar, May 12 2023

			1.51981775463506657...

G. C. Greubel, Table of n, a(n) for n = 1..1000
John Christopher, The Asymptotic Density of Some k-Dimensional Sets, The American Mathematical Monthly, Vol. 63, No. 6 (1956), pp. 399-401.
Werner Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Vol. 14, No. 2 (2015), pp. 73-88.
Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
Michael Penn, Irregular numbers and the coolest sum I have ever seen!, 2021 video.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., Vol. 5 (1913), pp. 105-106.
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link (p. 230).
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Moebius Function.
Eric Weisstein's World of Mathematics, Prime Products.
Index entries for transcendental numbers.

Cf. A001615, A004709, A005117, A007434, A013661, A013662, A157290.

SetDefaultRealField(RealField(100)); R:= RealField(); 15/Pi(R)^2; // G. C. Greubel, Oct 18 2019

evalf(15/Pi^2, 120); # G. C. Greubel, Oct 18 2019

A082020[digits_] := First[RealDigits[Zeta[2]/Zeta[4],10,digits]]; A082020[100] (* Enrique Pérez Herrero, Jan 15 2012 *)
RealDigits[15/Pi^2,10,120][[1]] (* Harvey P. Dale, Jun 23 2019 *)

PARI
```
15/Pi^2 \\ Michel Marcus, Oct 18 2019
```

numerical_approx(15/pi^2, digits=100) # G. C. Greubel, Oct 18 2019

Product_{n >= 1} (1+1/prime(n)^2) = 15/Pi^2 (Ramanujan).
Equals zeta(2)/zeta(4) = A013661/A013662 = Sum_{n>=1} mu(n)^2/n^2 = Sum_{n>=1} |mu(n)|/n^2 . - Enrique Pérez Herrero, Jan 15 2012
Equals Sum_{n>=1} 1/A005117(n)^2 . - Enrique Pérez Herrero, Mar 30 2012
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} psi(k)/k, where psi(k) is the Dedekind psi function (A001615). - Amiram Eldar, May 12 2019.
Equals Sum_{k>=1} A007434(k)/k^4. - Amiram Eldar, Jan 25 2024

A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

Sum_{n <= x} A189021(n) ~ kx, where k is this constant. - Charles R Greathouse IV, Jan 24 2018

The probability that a number chosen at random is squarefree and coprime to another randomly chosen random (see Schroeder, 2009). - Amiram Eldar, May 23 2020, corrected Aug 04 2020

			0.428249505677094440218765707581823546...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 110.
Manfred Schroeder, Number Theory in Science and Communication, 5th edition, Springer, 2009, page 59.

R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011; Equation (116).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A000010, A057434, A078078, A056671.
Cf. A065463, A013661.
Cf. A065473, A065480.

$MaxExtraPrecision = 800; digits = 98; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms+10]]; r[n_Integer] := LR[[n]]; (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n-1]/(n-1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - (2*p-1)/p^3) \\ Amiram Eldar, Mar 12 2021

Equals A065463 divided by A013661. - R. J. Mathar, Mar 22 2011
Equals A065473 divided by A065480. - R. J. Mathar, May 02 2019
Equals (6/Pi^2)^2 * Product_{p prime} (1 + 1/(p^3 + p^2 - p - 1)) = 1.1587609... * (6/Pi^2)^2. - Amiram Eldar, May 23 2020
Equals lim_{m->oo} (1/m) * Sum_{k==1..m} (phi(k)/k)^2, where phi is the Euler totient function (A000010). - Amiram Eldar, Mar 12 2021

More digits from Vaclav Kotesovec, Dec 18 2019

A111003 Decimal expansion of Pi^2/8.

Original entry on oeis.org

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

Offset: 1

Sam Alexander, Oct 01 2005

According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - Alonso del Arte, Apr 01 2013

Equals the asymptotic mean of the abundancy index of the odd numbers. - Amiram Eldar, May 12 2023

			1.23370055013616982735431137498451889191421242590509882830166867202...
1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017

F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
L. B. W. Jolley, Summation of Series, Dover (1961).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (5).
Index entries for transcendental numbers

Cf. A013661 (Pi^2/6), A002388, A102753, A091476, A309891.

RealDigits[Pi^2/8, 10, 105][[1]] (* Robert G. Wilson v *)

Pi^2/8 \\ Charles R Greathouse IV, Dec 04 2016

sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]
Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - Alonso del Arte, Aug 15 2012
Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - Alonso del Arte, Mar 29 2013
Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - Jean-François Alcover, Apr 29 2013
Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - Bruno Berselli, May 13 2013
Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [Jean-François Alcover, May 17 2013, after Boros & Moll p. 230]
Equals A002388/8 = A102753/4 = A091476/2. - R. J. Mathar, Oct 13 2015
Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - R. J. Mathar, Oct 22 2015
Equals (3/4)*zeta(2) = (3/4)*A013661. - Wolfdieter Lang, Sep 02 2019
From Amiram Eldar, Jul 17 2020: (Start)
Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.
Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.
Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)
Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{n>=1} A309891(n)/n^2. - Friedjof Tellkamp, Jan 25 2025
Equals lambda(2), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

More terms from Robert G. Wilson v, Oct 04 2005

A260832 a(n) = numerator(Jtilde2(n)).

Original entry on oeis.org

1, 3, 41, 147, 8649, 32307, 487889, 1856307, 454689481, 1748274987, 26989009929, 104482114467, 6488426222001, 25239009088827, 393449178700161, 1535897056631667, 1537112996582116041, 6016831929058214523, 94316599529950360769, 369994845516850143483, 23244865440911268112681

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde2(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(2), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..830
Takashi Ichinose and Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto and Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 1).
Ling Long, Robert Osburn and Holly Swisher, On a conjecture of Kimoto and Wakayama, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.

Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
seq(a(n), n = 0..20); # Peter Luschny, Dec 08 2022

Numerator[Table[Sum[ (-1)^k*Binomial[-1/2, k]^2*Binomial[n, k], {k, 0, n}], {n,0,50}]] (* G. C. Greubel, Feb 15 2017 *)

a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2,k)^2*binomial(n, k)));

a(n) = numerator(sum(k=0, n, binomial(2*k, k)*binomial(4*k, 2*k)* binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))) / (2^(4*n)* binomial(2*n,n)));

Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
And 4n^2*J2(n) - (8n^2-8n+3)*J2(n-1) + 4(n-1)^2*J2(n-2) = 0 with J2(0) = 3*zeta(2) and J2(1) = 9*zeta(2)/4.
Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
Jtilde2(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(4*k,2*k)*binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))/(2^(4*n)*binomial(2*n,n)).
From Andrey Zabolotskiy, Oct 04 2016 and Dec 08 2022: (Start)
Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
Jtilde2(n) = A143583(n) / 16^n. (End)
a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - Peter Luschny, Dec 08 2022

A084920 a(n) = (prime(n)-1)*(prime(n)+1).

Original entry on oeis.org

3, 8, 24, 48, 120, 168, 288, 360, 528, 840, 960, 1368, 1680, 1848, 2208, 2808, 3480, 3720, 4488, 5040, 5328, 6240, 6888, 7920, 9408, 10200, 10608, 11448, 11880, 12768, 16128, 17160, 18768, 19320, 22200, 22800, 24648, 26568, 27888, 29928

Offset: 1

Reinhard Zumkeller, Jun 11 2003

Squares of primes minus 1. - Wesley Ivan Hurt, Oct 11 2013

Integers k for which there exist exactly two positive integers b such that (k+1)/(b+1) is an integer. - Benedict W. J. Irwin, Jul 26 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 6.

Cf. A000040, A005563, A049001, A154945, A166010, A182200, A182174.
Cf. A006093, A008864, A084921, A084922.
Cf. A066872, A001248, A024702, A013661, A065469.

a084920 n = (p - 1) * (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Aug 27 2013

[p^2-1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 30 2015

A084920:=n->ithprime(n)^2-1; seq(A084920(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2 - 1, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)
Prime[Range[50]]^2-1 (* Harvey P. Dale, Oct 02 2021 *)

a(n) = (prime(n)-1)*(prime(n)+1); \\ Michel Marcus, Jul 28 2016

[(p-1)*(p+1) for p in primes(200)] # Bruno Berselli, Mar 30 2015

a(n) = A006093(n) * A008864(n);
a(n) = A084921(n)*2, for n > 1; a(n) = A084922(n)*6, for n > 2.
Product_{n > 0} a(n)/A066872(n) = 2/5. a(n) = A001248(n) - 1. - R. J. Mathar, Feb 01 2009
a(n) = prime(n)^2 - 1 = A001248(n) - 1. - Vladimir Joseph Stephan Orlovsky, Oct 17 2009
a(n) ~ n^2*log(n)^2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^2*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime. - Seiichi Manyama, Dec 31 2017
a(n) = 24 * A024702(n) for n > 2. - Jianing Song, Apr 28 2019
Sum_{n>=1} 1/a(n) = A154945. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = Pi^2/6 (A013661).
Product_{n>=1} (1 - 1/a(n)) = A065469. (End)

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

sign

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n).

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 13, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 13, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 22, 1, 4, 7, 6, 4, 13, 1, 7, 4, 13, 1, 17, 1, 4, 7, 7, 4, 13, 1, 13, 4, 4, 1, 22, 4, 4, 4, 10, 1, 22, 4, 7, 4, 4, 4

Offset: 1

Henry Bottomley, Jul 23 2001

Also number of ways to write 1/n as sum of exactly two distinct unit fractions. - Thomas L. York, Jan 11 2014
Also number of positive integers m such that 1/n + 1/m is a unit fraction. - Jon E. Schoenfield, Apr 17 2018
If 1/n = 1/b - 1/c then n = bc/(c-b) and 1/n = 1/(2n-b) + 1/(c+2n) (though it is also the case that 1/n = 1/(2n) + 1/(2n) equivalent to b = c = 0).
Also number of divisors of n^2 less than n. - Vladeta Jovovic, Aug 13 2001
Number of elements in the set {(x,y): x|n, y|n, xVladeta Jovovic, May 03 2002
Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre, Jan 04 2002
This is similar to A062799, having the same first 29 terms. But they are different sequences.
If A001221(n) = omega(n) <= 2, then a(n) = A062799(n); if A001221(n) > 2, then a(n) > A062799(n). - Matthew Vandermast, Aug 25 2004
Number of r X s integer-sided rectangles such that r + s = 4n, r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020
Also number of integer-sided right triangles with 2n as a leg. Equivalent to the even indices of A046079. - Nathaniel C Beckman, May 14 2020; Jun 26 2020
a(n) is the number of positive integers k such that k+n divides k*n. - Thomas Ordowski, Dec 02 2024

			a(10) = 4 since 1/10 = 1/5 - 1/10 = 1/6 - 1/15 = 1/8 - 1/40 = 1/9 - 1/90.
a(12) = 7: the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the decompositions are (1, 2), (1, 3), (1, 4), (1, 6), (1, 12), (2, 3), (3, 4).

T. D. Noe, Table of n, a(n) for n = 1..10000
Christopher J. Bradley, Solution to Problem 2175, Crux Mathematicorum, Vol. 23, No. 7, (Nov 1997), pp. 443-444.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Roger B. Eggleton, Unitary Fractions: 10501, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 372.
Amiram Eldar, Plot of (Sum_{i=1..n} a(i))/f(n) - 1, where f(n) is an asymptotic function, for n = 2^(1..28).
Amiram Eldar, Plot of Sum_{i=1..n} a(i)/(n*log(n)*log(log(n))) for n = 2^(1..28).
Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001, pp. 303-306.

Cf. A018892, A063427, A063428, A129296.
First twenty-nine terms identical to those of A062799.
Cf. A001221, A046079, A048691, A063717, A063718.
Cf. A001620, A082633, A013661, A201994, A306016.

[(NumberOfDivisors(n^2)-1)/2 : n in [1..100]]; // Vincenzo Librandi, Apr 18 2018

Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
(DivisorSigma[0,Range[100]^2]-1)/2 (* Harvey P. Dale, Apr 15 2013 *)

for(n=1,100,print1(sum(i=1,n^2,if((n*i)%(i+n),0,1)),","))

a(n)=numdiv(n^2)\2 \\ Charles R Greathouse IV, Oct 03 2016

a(n) = (tau(n^2)-1)/2.
a(n) = A018892(n)-1. If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1+1)(2*a2+1)...(2*at+1)-1)/2.
If n is prime a(n)=1. Conjecture: (1/n)*Sum_{i=1..n} a(i) = C*log(n)*log(log(n)) + o(log(n)) with C=0.7... [The conjecture is false. See the plot and the asymptotic formula below. - Amiram Eldar, Oct 03 2024]
Bisection of A046079. - Lekraj Beedassy, Jul 09 2004
a(n) = Sum_{i=1..2*n-1} (1 - ceiling(i*(4*n-i)/(4*n-2*i)) + floor(i*(4*n-i)/(4*n-2*i))). - Wesley Ivan Hurt, Apr 24 2020
Sum_{k=1..n} a(k) ~ (n/(2*zeta(2)))*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 - zeta(2) + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Oct 03 2024

A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

From Richard R. Forberg, May 22 2023: (Start)
This constant is the asymptotic mean of (phi(n)/n)*(sigma(n)/n), where phi is the Euler totient function (A000010) and sigma is the sum-of-divisors function (A000203).
In contrast, the product of the separate means, mean(phi(n)/n) * mean(sigma(n)/n), converges to 1, with the asymptotic mean(sigma(n)/n) = Pi^2/6 = zeta(2). See A013661.
Also see A062354. (End)

			0.88151383972517077692839182290...

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Steven R. Finch, Class number theory, page 7. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50 and 85.
R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Simon Plouffe, Generalized expansions of real numbers, 2006.
Eric Weisstein's World of Mathematics, Quadratic Class Number Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A007434, A013661, A062354, A078087.

$MaxExtraPrecision = 1000; digits = 98; terms = 1000; LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*PrimeZetaP[n-1]/(n-1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Mar 14 2021

Sum_{n>=1} phi(n)/(n*J(n)) = (this constant)*A013661 with phi()=A000010() and J() = A007434() [Cohen, Corollary 5.1.1]. - R. J. Mathar, Apr 11 2011

A175646 Decimal expansion of the Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 01 2010

The Euler product of the Riemann zeta function at 2 restricted to primes in A002476, which is the inverse of the infinite product (1-1/7^2)*(1-1/13^2)*(1-1/19^2)*...
There is a complementary Product_{primes p == 2 (mod 3)} 1/(1-1/p^2) = A333240 = 1.4140643908921476375655018190798... such that (this constant here)*1.4140643.../(1-1/3^2) = zeta(2) = A013661.
Because 1/(1-p^(-2)) = 1+1/(p^2-1), the complementary 1.414064... also equals Product_{primes p == 2 (mod 3)} (1+1/(p^2-1)), which appears in Eq. (1.8) of [Dence and Pomerance]. - R. J. Mathar, Jan 31 2013

			1.03401487541434188053903064441304762857896...

Peter Luschny, Table of n, a(n) for n = 1..1000 (terms 1..105 from Vaclav Kotesovec).
Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, alternative link.
S. Ettahri, O. Ramare, L. Surel, Fast multi-precision computation of some Euler products, arxiv:1908.06808 (2019), Section 9.
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, p. 26.

Cf. A002476, A004611, A007528, A175647, A301429, A333240, A334478, A334480.

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n) / 3:
evalf(4*Pi^2 / (27*mul(x(n), n=1..8)), 106); # Peter Luschny, Jan 17 2021

digits = 105;
precision = digits + 5;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pB = (3/4)*Product[gv[2^n*2]^(2^-(n+1)), {n, 0, 11}] // N[#, precision]&;
pA = Pi^2/9/pB ;
RealDigits[pA, 10, digits][[1]]
(* Jean-François Alcover, Jan 11 2021, after PARI code due to Artur Jasinski *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[3,1,2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n] / Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n) / 3;
N[4 Pi^2 / (27 Product[x[n], {n, 8}]), 106] (* Peter Luschny, Jan 17 2021 *)

Equals 2*Pi^2 / (3^(7/2) * A301429^2). - Vaclav Kotesovec, May 12 2020

Equals Sum_{k>=1} 1/A004611(k)^2. - Amiram Eldar, Sep 27 2020

More digits from Vaclav Kotesovec, May 12 2020 and Jun 27 2020

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