A104777 -id:A104777 - cp's OEIS frontend

A100044 Decimal expansion of Pi^2/9.

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

Offset: 1

Eric W. Weisstein, Oct 31 2004

The Dirichlet L-series for the principal character mod 6 (which is A120325 shifted left) evaluated at 2. - R. J. Mathar, Jul 20 2012

Equals the asymptotic mean of the abundancy index of the numbers coprime to 6 (A007310). - Amiram Eldar, May 12 2023

			1.096622711232150957648276777764...

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.
L. B. W. Jolley, Summation of Series, Dover, 1961.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
Eric Weisstein's World of Mathematics, Digit Sum.
Index entries for transcendental numbers.

Cf. A000120, A007310, A008833, A019670, A086463, A091999, A104777, A120325, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/9, 10, 110][[1]] (* G. C. Greubel, Feb 17 2017 *)

default(realprecision, 110); Pi^2/9 \\ G. C. Greubel, Feb 17 2017

numerical_approx(pi^2/9, digits=120) # G. C. Greubel, Jun 02 2021

Equals 1 + (1/2)*(1/3)*(1/2) + (1/3)*(1*2)/(3*5)*(1/2)^2 + (1/4) *(1*2*3)/(3*5*7)*(1/2)^3 + .... [Jolley eq 277]
Equals 1/1^2 + 1/5^2 + 1/7^2 + 1/11^2 + 1/13^2 + 1/17^2 + .... - R. J. Mathar, Jul 20 2012
Equals 2*Sum_{n>=1} 1/(6*n*(3*n + (-1)^n - 3) - 3*(-1)^n + 5) = 2*Sum_{n>=1} 1/(2*A104777(n)). - Alexander R. Povolotsky, May 18 2014
Equals A019670^2. - Michel Marcus, May 19 2014
Equals 2*A086463 = 2*Sum_{n>=1} 1/A091999(n)^2, equivalent to the formula of 2012 above. - Alexander R. Povolotsky, May 20 2014
Equals 3F2(1,1,1; 3/2,2 ; 1/4), following from Clausen's formula of J. Reine Angew. Math 3 (1828) for squares of 2F1() as noted in A019670. - R. J. Mathar, Oct 16 2015
Equals Product_{n >= 3} prime(n)^2 / (prime(n)^2 - 1), Euler's prime product, excluding first two primes. - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=0..oo} log(x)/(x^6 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} A000120(k) * (2*k+1)/(k^2*(k+1)^2) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
Equals Integral_{x=0..1} log(1+x+x^2)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{k>=1} A008833(k)/k^4. - Amiram Eldar, Jan 25 2024
Continued fraction expansion: 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n-1)*n^3/((5*n^2+6*n+2) - ... ))))). See A130549. - Peter Bala, Feb 16 2024
Equals Sum_{k >= 0} 1/((k + 1)*(2*k + 1)*binomial(2*k, k)). See Catalan, Section 21, equation 30. - Peter Bala, Aug 14 2024

A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

			G.f. = x + x^25 + x^49 + x^121 + x^169 + x^289 + x^361 + x^529 + x^625 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 105, Eq. (41).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions.

Cf. A098108, A033684.

Cf. A010052, A010872, A104777.

a033683 n = fromEnum $ odd n && mod n 3 > 0 && a010052 n == 1
-- Reinhard Zumkeller, Nov 14 2015

Basis( ModularForms( Gamma0(144), 1/2), 106)[2]; /* Michael Somos, Dec 07 2019 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x^4] - EllipticTheta[ 2, 0, x^36])/2, {x, 0, n}] // PowerExpand; (* Michael Somos, Dec 07 2019 *)
Table[If[OddQ[n]&&IntegerQ[Sqrt[n]]&&Mod[n,3]!=0,1,0],{n,0,120}] (* Harvey P. Dale, Sep 06 2020 *)

{a(n) = if( n%24 == 1, issquare(n), 0)}; /* Michael Somos, Jan 26 2008 */

Essentially the series psi_6(z)=(1/2)(theta_2(z/9)-theta_2(z)).
a(A104777(n)) = 1.
A080995(n) = a(24n+1).
Multiplicative with a(p^e) = 1 if 2 divides e and p > 3, 0 otherwise. - Mitch Harris, Jun 09 2005
Euler transform of a period 144 sequence. - Michael Somos, Jan 26 2008
a(n) = A033684(n) * A000035(n).
Dirichlet g.f.: zeta(2*s) *(1-2^(-2s)) *(1-3^(-2s)). - R. J. Mathar, Mar 10 2011
G.f.: Sum_{k in Z} x^(6*k+1)^2. - Michael Somos, Dec 07 2019
Sum_{k=1..n} a(k) ~ sqrt(n)/3. - Amiram Eldar, Jan 14 2024

A204999 a(n) = (1/n)*A204998(n).

Original entry on oeis.org

3, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Positions of 3's seem to be given by a subsequence of A104777. - Antti Karttunen, Sep 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..23005

Cf. A204996, A204997, A204998, A204892.

Mathematica
```
(See the program at A204994.)
```

A204999(n) = { my(d); for(k=sqrtint(1+n), oo, for(j=1,k-1,if(!((d=(k^2)-(j^2))%n),return(d/n),if(dAntti Karttunen, Sep 28 2018

a(n) = (A204996(n)-A204997(n))/n.

More terms from Antti Karttunen, Sep 28 2018

A164097 Numbers k such that 6*k + 7 is a perfect square.

Original entry on oeis.org

3, 7, 19, 27, 47, 59, 87, 103, 139, 159, 203, 227, 279, 307, 367, 399, 467, 503, 579, 619, 703, 747, 839, 887, 987, 1039, 1147, 1203, 1319, 1379, 1503, 1567, 1699, 1767, 1907, 1979, 2127, 2203, 2359, 2439, 2603, 2687, 2859, 2947, 3127, 3219, 3407, 3503, 3699

Offset: 1

Vincenzo Librandi, Aug 10 2009

The entries are prime, or divisible by 3, or divisible by prime of the form 3*m+1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A062717, A104777 (the squares 6*k+7).

[n: n in [1..4000] | IsSquare(6*n+7)]; // Vincenzo Librandi, Oct 12 2012

Select[Range[4000], IntegerQ[Sqrt[6 # + 7 ]] &] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {3, 7, 19, 27, 47}, 50] (* Harvey P. Dale, Apr 29 2011 *)

From R. J. Mathar, Aug 26 2009: (Start)
a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
G.f.: x*(-3-4*x-6*x^2+x^4)/((1+x)^2*(x-1)^3).
a(n) = 3*(2*n-1+2*n^2)/4 -(-1)^n*(1+2*n)/4 = A062717(n+1)-1. (End)
Sum_{n>=1} 1/a(n) = 1 + (tan((2+sqrt(7))*Pi/6) - cot((1+sqrt(7))*Pi/6))*Pi/(2*sqrt(7)). - Amiram Eldar, Feb 24 2023

Edited by R. J. Mathar, Aug 26 2009

A229852 3*h^2, where h is an odd integer not divisible by 3.

Original entry on oeis.org

3, 75, 147, 363, 507, 867, 1083, 1587, 1875, 2523, 2883, 3675, 4107, 5043, 5547, 6627, 7203, 8427, 9075, 10443, 11163, 12675, 13467, 15123, 15987, 17787, 18723, 20667, 21675, 23763, 24843, 27075, 28227, 30603, 31827, 34347, 35643, 38307, 39675, 42483, 43923

Offset: 1

Arkadiusz Wesolowski, Oct 01 2013

If p = a(n)*2^k + 1 divides a composite Fermat number 2^(2^m) + 1 and p is a prime, then k is odd.

More precisely, k == 1 (mod 4) if h == +/- 1 (mod 5) and k == 3 (mod 4) if h == +/- 2 (mod 5) (Krizek, Luca and Somer).

M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, pp. 63-65.

Colin Barker, Table of n, a(n) for n = 1..1000
Wilfrid Keller, Fermat factoring status.
Eric Weisstein's World of Mathematics, Fermat Number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000215, A204620, A291050.

[3*h^2 : h in [1..121 by 2] | not IsZero(h mod 3)];

3*Select[Range[1, 121, 2], Mod[#, 3] > 0 &]^2 (* Amiram Eldar, Jan 02 2021 *)

forstep(h=1, 121, 2, if(!(h%3==0), print1(3*h^2, ", ")));

Vec(3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

G.f.: 3*x*(1+24*x+22*x^2+24*x^3+x^4) / ((1-x)^3*(1+x)^2).
a(n) = 3*A104777(n).
From Colin Barker, Jan 26 2016: (Start)
a(n) = 3*(18*n^2+6*(-1)^n*n-18*n-3*(-1)^n+5)/2.
a(n) = 27*n^2-18*n+3 for n even.
a(n) = 27*n^2-36*n+12 for n odd.
(End)
Sum_{n>=1} 1/a(n) = Pi^2/27 (A291050). - Amiram Eldar, Jan 02 2021

cp's OEIS Frontend

A100044 Decimal expansion of Pi^2/9.

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A033683 a(n) = 1 if n is an odd square not divisible by 3, otherwise 0.

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A204999 a(n) = (1/n)*A204998(n).

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A164097 Numbers k such that 6*k + 7 is a perfect square.

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Programs

Magma

Mathematica

Formula

Extensions

A229852 3*h^2, where h is an odd integer not divisible by 3.

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Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula