A002194 -id:A002194 - cp's OEIS frontend

A007310 Numbers congruent to 1 or 5 mod 6.

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175

Offset: 1

C. Christofferson (Magpie56(AT)aol.com)

Numbers n such that phi(4n) = phi(3n). - Benoit Cloitre, Aug 06 2003
Or, numbers relatively prime to 2 and 3, or coprime to 6, or having only prime factors >= 5; also known as 5-rough numbers. (Edited by M. F. Hasler, Nov 01 2014: merged with comments from Zak Seidov, Apr 26 2007 and Michael B. Porter, Oct 09 2009)
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 38 ).
Numbers k such that k mod 2 = 1 and (k+1) mod 3 <> 1. - Klaus Brockhaus, Jun 15 2004
Also numbers n such that the sum of the squares of the first n integers is divisible by n, or A000330(n) = n*(n+1)*(2*n+1)/6 is divisible by n. - Alexander Adamchuk, Jan 04 2007
Numbers n such that the sum of squares of n consecutive integers is divisible by n, because A000330(m+n) - A000330(m) = n*(n+1)*(2*n+1)/6 + n*(m^2+n*m+m) is divisible by n independent of m. - Kaupo Palo, Dec 10 2016
A126759(a(n)) = n + 1. - Reinhard Zumkeller, Jun 16 2008
Terms of this sequence (starting from the second term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008
For n > 1: a(n) is prime if and only if A075743(n-2) = 1; a(2*n-1) = A016969(n-1), a(2*n) = A016921(n-1). - Reinhard Zumkeller, Oct 02 2008
A156543 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Numbers n such that ChebyshevT(x, x/2) is not an integer (is integer/2). - Artur Jasinski, Feb 13 2010
If 12*k + 1 is a perfect square (k = 0, 2, 4, 10, 14, 24, 30, 44, ... = A152749) then the square root of 12*k + 1 = a(n). - Gary Detlefs, Feb 22 2010
A089128(a(n)) = 1. Complement of A047229(n+1) for n >= 1. See A164576 for corresponding values A175485(a(n)). - Jaroslav Krizek, May 28 2010
Cf. property described by Gary Detlefs in A113801 and in Comment: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (with h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 6). Also a(n)^2 - 1 == 0 (mod 12). - Bruno Berselli, Nov 05 2010 - Nov 17 2010
Numbers n such that ( Sum_{k = 1..n} k^14 ) mod n = 0. (Conjectured) - Gary Detlefs, Dec 27 2011
From Peter Bala, May 02 2018: (Start)
The above conjecture is true. Apply Ireland and Rosen, Proposition 15.2.2. with m = 14 to obtain the congruence 6*( Sum_{k = 1..n} k^14 )/n = 7 (mod n), true for all n >= 1.  Suppose n is coprime to 6, then 6 is a unit in Z/nZ, and it follows from the congruence that ( Sum_{k = 1..n} k^14 )/n is an integer. On the other hand, if either 2 divides n or 3 divides n then the congruence shows that ( Sum_{k = 1..n} k^14 )/n cannot be integral. (End)
A126759(a(n)) = n and A126759(m) < n for m < a(n). - Reinhard Zumkeller, May 23 2013
(a(n-1)^2 - 1)/24 = A001318(n), the generalized pentagonal numbers. - Richard R. Forberg, May 30 2013
Numbers k for which A001580(k) is divisible by 3. - Bruno Berselli, Jun 18 2014
Numbers n such that sigma(n) + sigma(2n) = sigma(3n). - Jahangeer Kholdi and Farideh Firoozbakht, Aug 15 2014
a(n) are values of k such that Sum_{m = 1..k-1} m*(k-m)/k is an integer. Sums for those k are given by A062717. Also see Detlefs formula below based on A062717. - Richard R. Forberg, Feb 16 2015
a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)/6 is integral, and also such that the sequence c(n) = n*(n + m)*(n + 2*m)*(n + 3*m)/24 is integral. Cf. A007775. - Peter Bala, Nov 13 2015
Along with 2, these are the numbers k such that the k-th Fibonacci number is coprime to every Lucas number. - Clark Kimberling, Jun 21 2016
This sequence is the Engel expansion of 1F2(1; 5/6, 7/6; 1/36) + 1F2(1; 7/6, 11/6; 1/36)/5. - Benedict W. J. Irwin, Dec 16 2016
The sequence a(n), n >= 4 is generated by the successor of the pair of polygonal numbers {P_s(4) + 1, P_(2*s - 1)(3) + 1}, s >= 3. - Ralf Steiner, May 25 2018
The asymptotic density of this sequence is 1/3. - Amiram Eldar, Oct 18 2020
Also, the only vertices in the odd Collatz tree A088975 that are branch values to other odd nodes t == 1 (mod 2) of A005408. - Heinz Ebert, Apr 14 2021
From Flávio V. Fernandes, Aug 01 2021: (Start)
For any two terms j and k, the product j*k is also a term (the same property as p^n and smooth numbers).
From a(2) to a(phi(A033845(n))), or a((A033845(n))/3), the terms are the totatives of the A033845(n) itself. (End)
Also orders n for which cyclic and semicyclic diagonal Latin squares exist (see A123565 and A342990). - Eduard I. Vatutin, Jul 11 2023
If k is in the sequence, then k*2^m + 3 is also in the sequence, for all m > 0. - Jules Beauchamp, Aug 29 2024

			G.f. = x + 5*x^2 + 7*x^3 + 11*x^4 + 13*x^5 + 17*x^6 + 19*x^7 + 23*x^8 + ...

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1980.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.
B. W. J. Irwin, Constants Whose Engel Expansions are the k-rough Numbers.
L. B. W. Jolley, Summation of Series, Dover, 1961
Cedric A. B. Smith, Prime factors and recurring duodecimals, Math. Gaz. 59 (408) (1975) 106-109.
William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N).
Eric Weisstein's World of Mathematics, Rough Number.
Eric Weisstein's World of Mathematics, Pi Formulas. [_Jaume Oliver Lafont_, Oct 23 2009]
Index entries for sequences related to smooth numbers [_Michael B. Porter_, Oct 09 2009]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

A005408 \ A016945. Union of A016921 and A016969; union of A038509 and A140475. Essentially the same as A038179. Complement of A047229. Subsequence of A186422.
Cf. A000330, A001580, A002194, A019670, A032528 (partial sums), A038509 (subsequence of composites), A047209, A047336, A047522, A056020, A084967, A090771, A091998, A144065, A175885-A175887.
For k-rough numbers with other values of k, see A000027, A005408, A007775, A008364-A008366, A166061, A166063.
Cf. A126760 (a left inverse).
Row 3 of A260717 (without the initial 1).
Cf. A105397 (first differences).

Filtered([1..150],n->n mod 6=1 or n mod 6=5); # Muniru A Asiru, Dec 19 2018

a007310 n = a007310_list !! (n-1)
a007310_list = 1 : 5 : map (+ 6) a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..250] | n mod 6 in [1, 5]]; // Vincenzo Librandi, Feb 12 2016

seq(seq(6*i+j,j=[1,5]),i=0..100); # Robert Israel, Sep 08 2014

Select[Range[200], MemberQ[{1, 5}, Mod[#, 6]] &] (* Harvey P. Dale, Aug 27 2013 *)
a[n_] := (6 n + (-1)^n - 3)/2; a[rem156, 60] (* Robert G. Wilson v, May 26 2014 from a suggestion by N. J. A. Sloane *)
Flatten[Table[6n + {1, 5}, {n, 0, 24}]] (* Alonso del Arte, Feb 06 2016 *)
Table[2*Floor[3*n/2] - 1, {n, 1000}] (* Mikk Heidemaa, Feb 11 2016 *)

isA007310(n) = gcd(n,6)==1 \\ Michael B. Porter, Oct 09 2009

A007310(n)=n\2*6-(-1)^n \\ M. F. Hasler, Oct 31 2014

\\ given an element from the sequence, find the next term in the sequence.
nxt(n) = n + 9/2 - (n%6)/2 \\ David A. Corneth, Nov 01 2016

def A007310(n): return (n+(n>>1)<<1)-1 # Chai Wah Wu, Oct 10 2023

[i for i in range(150) if gcd(6,i) == 1] # Zerinvary Lajos, Apr 21 2009

a(n) = (6*n + (-1)^n - 3)/2. - Antonio Esposito, Jan 18 2002
a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4. - Roger L. Bagula
a(n) = 3*n - 1 - (n mod 2). - Zak Seidov, Jan 18 2006
a(1) = 1 then alternatively add 4 and 2. a(1) = 1, a(n) = a(n-1) + 3 + (-1)^n. - Zak Seidov, Mar 25 2006
1 + 1/5^2 + 1/7^2 + 1/11^2 + ... = Pi^2/9 [Jolley]. - Gary W. Adamson, Dec 20 2006
For n >= 3 a(n) = a(n-2) + 6. - Zak Seidov, Apr 18 2007
From R. J. Mathar, May 23 2008: (Start)
Expand (x+x^5)/(1-x^6) = x + x^5 + x^7 + x^11 + x^13 + ...
O.g.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^2). (End)
a(n) = 6*floor(n/2) - 1 + 2*(n mod 2). - Reinhard Zumkeller, Oct 02 2008
1 + 1/5 - 1/7 - 1/11 + + - - ... = Pi/3 = A019670 [Jolley eq (315)]. - Jaume Oliver Lafont, Oct 23 2009
a(n) = ( 6*A062717(n)+1 )^(1/2). - Gary Detlefs, Feb 22 2010
a(n) = 6*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), with n > 1. - Bruno Berselli, Nov 05 2010
a(n) = 6*n - a(n-1) - 6 for n>1, a(1) = 1. - Vincenzo Librandi, Nov 18 2010
Sum_{n >= 1} (-1)^(n+1)/a(n) = A093766 [Jolley eq (84)]. - R. J. Mathar, Mar 24 2011
a(n) = 6*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
a(n) = 3*n + ((n+1) mod 2) - 2. - Gary Detlefs, Jan 08 2012
a(n) = 2*n + 1 + 2*floor((n-2)/2) = 2*n - 1 + 2*floor(n/2), leading to the o.g.f. given by R. J. Mathar above. - Wolfdieter Lang, Jan 20 2012
1 - 1/5 + 1/7 - 1/11 + - ... = Pi*sqrt(3)/6 = A093766 (L. Euler). - Philippe Deléham, Mar 09 2013
1 - 1/5^3 + 1/7^3 - 1/11^3 + - ... = Pi^3*sqrt(3)/54 (L. Euler). - Philippe Deléham, Mar 09 2013
gcd(a(n), 6) = 1. - Reinhard Zumkeller, Nov 14 2013
a(n) = sqrt(6*n*(3*n + (-1)^n - 3)-3*(-1)^n + 5)/sqrt(2). - Alexander R. Povolotsky, May 16 2014
a(n) = 3*n + 6/(9*n mod 6 - 6). - Mikk Heidemaa, Feb 05 2016
From Mikk Heidemaa, Feb 11 2016: (Start)
a(n) = 2*floor(3*n/2) - 1.
a(n) = A047238(n+1) - 1. (suggested by Michel Marcus) (End)
E.g.f.: (2 + (6*x - 3)*exp(x) + exp(-x))/2. - Ilya Gutkovskiy, Jun 18 2016
From Bruno Berselli, Apr 27 2017: (Start)
a(k*n) = k*a(n) + (4*k + (-1)^k - 3)/2 for k>0 and odd n, a(k*n) = k*a(n) + k - 1 for even n. Some special cases:
k=2: a(2*n) = 2*a(n) + 3 for odd n, a(2*n) = 2*a(n) + 1 for even n;
k=3: a(3*n) = 3*a(n) + 4 for odd n, a(3*n) = 3*a(n) + 2 for even n;
k=4: a(4*n) = 4*a(n) + 7 for odd n, a(4*n) = 4*a(n) + 3 for even n;
k=5: a(5*n) = 5*a(n) + 8 for odd n, a(5*n) = 5*a(n) + 4 for even n, etc. (End)
From Antti Karttunen, May 20 2017: (Start)
a(A273669(n)) = 5*a(n) = A084967(n).
a((5*n)-3) = A255413(n).
A126760(a(n)) = n. (End)
a(2*m) = 6*m - 1, m >= 1; a(2*m + 1) = 6*m + 1, m >= 0. - Ralf Steiner, May 17 2018
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(3) (A002194).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi/3 (A019670). (End)

A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 12 2011

The Mathematica program includes a graph.
Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:
b.....c.......x
1.....2.......A197133
1.....3.......A197134
1.....4.......A197135
1.....5.......A197251
1.....6.......A197252
1.....7.......A197253
1.....8.......A197254
2.....1.......A105199, x=arctan(2)
2.....3.......A019679, x=Pi/12
2.....4.......A197255
2.....5.......A197256
2.....6.......A197257
2.....7.......A197258
2.....8.......A197259
3.....1.......A197260
3.....2.......A197261
3.....4.......A197262
3.....5.......A197263
3.....6.......A197264
3.....7.......A197265
3.....8.......A197266
4.....1.......A197267
4.....2.......A195693, x=arctan(1/(golden ratio))
4.....3.......A197268
1.....4*Pi....A197522
1.....3*Pi....A197571
1.....2*Pi....A197572
1.....3*Pi/2..A197573
1.....Pi......A197574
1.....Pi/2....A197575
1.....Pi/3....A197326
1.....Pi/4....A197327
1.....Pi/6....A197328
2.....Pi/3....A197329
2.....Pi/4....A197330
2.....Pi/6....A197331
3.....Pi/3....A197332
3.....Pi/6....A197375
3.....Pi/4....A197333
1.....1/2.....A197376
1.....1/3.....A197377
1.....2/3.....A197378
Pi....1.......A197576
Pi....2.......A197577
Pi....3.......A197578
2*Pi..1.......A197585
3*Pi..1.......A197586
4*Pi..1.......A197587
Pi/2..1.......A197579
Pi/2..2.......A197580
Pi/2..1/2.....A197581
Pi/3..Pi/4....A197379
Pi/3..Pi/6....A197380
Pi/4..Pi/3....A197381
Pi/4..Pi/6....A197382
Pi/6..Pi/3....A197383
Pi/6..Pi/4..........., x=1
Pi/3..1.......A197384
Pi/3..2.......A197385
Pi/3..3.......A197386
Pi/3..1/2.....A197387
Pi/3..1/3.....A197388
Pi/3..2/3.....A197389
Pi/4..1.......A197390
Pi/4..2.......A197391
Pi/4..3.......A197392
Pi/4..1/2.....A197393
Pi/4..1/3.....A197394
Pi/4..2/3.....A197411
Pi/4..1/4.....A197412
Pi/6..1.......A197413
Pi/6..2.......A197414
Pi/6..3.......A197415
Pi/6..1/2.....A197416
Pi/6..1/3.....A197417
Pi/6..2/3.....A197418
Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.

			0.272971849236824950408616...

Index entries for transcendental numbers.

Cf. A002194, A197134, A197476 (cos), A333322.

b = 1; c = 2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t] (* A197133 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
(* Second program: *)
RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)

asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ Gleb Koloskov, Sep 15 2021

asin(polrootsreal(4*x^3-4*x+1)[2]) \\ Charles R Greathouse IV, Feb 12 2025

From Gleb Koloskov, Sep 15 2021: (Start)
Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))
     = arcsin(2*sin(arcsin(A333322)/3)/A002194). (End)

Edited and a(99) corrected by Georg Fischer, Jul 28 2021

A010527 Decimal expansion of sqrt(3)/2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This is the ratio of the height of an equilateral triangle to its base.
Essentially the same sequence arises from decimal expansion of square root of 75, which is 8.6602540378443864676372317...
Also the real part of i^(1/3), the cubic root of i. - Stanislav Sykora, Apr 25 2012
Gilbert & Pollak conjectured that this is the Steiner ratio rho_2, the least upper bound of the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 2. (See Ivanov & Tuzhilin for the status of this conjecture as of 2012.) - Charles R Greathouse IV, Dec 11 2012
Surface area of a regular icosahedron with unit edge is 5*sqrt(3), i.e., 10 times this constant. - Stanislav Sykora, Nov 29 2013
Circumscribed sphere radius for a cube with unit edges. - Stanislav Sykora, Feb 10 2014
Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - Enrique Pérez Herrero, Nov 13 2014
Area of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016
If a, b, c are the sides of a triangle ABC and h_a, h_b, h_c the corresponding altitudes, then (h_a+h_b+h_c) / (a+b+c) <= sqrt(3)/2; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 26 2022

			0.86602540378443864676372317...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.2, 8.3 and 8.6, pp. 484, 489, and 504.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), pp. 450-451.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, and J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.8, page 114.

Harry J. Smith, Table of n, a(n) for n = 0..20000
E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16, (1968), pp. 1-29.
A. O. Ivanov and A. A. Tuzhilin, The Steiner ratio Gilbert-Pollak conjecture is still open, Algorithmica 62:1-2 (2012), pp. 630-632.
Matt Parker, The mystery of 0.866025403784438646763723170752936183471402626905190314027903489, Stand-up Maths, YouTube video, Feb 14 2024.
Simon Plouffe, Plouffe's Inverter, sqrt(3)/2 to 10000 digits.
Simon Plouffe, Sqrt(3)/2 to 5000 digits.
Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem.
Wikipedia, Icosahedron.
Wikipedia, Platonic solid.
Wikipedia, Unified Thread Standard.
Index entries for algebraic numbers, degree 2.

Cf. A010153.
Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron).
Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Cf. A126664 (continued fraction), A144535/A144536 (convergents).
Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).

SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[Sqrt[3]/2, 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=10*(sqrt(3)/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010527.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

sqrt(3)/2 \\ Michel Marcus, Apr 10 2016

Equals cos(30 degrees). - Kausthub Gudipati, Aug 15 2011
Equals A002194/2. - Stanislav Sykora, Nov 30 2013
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals 1/(10*A020832). - Bernard Schott, Sep 29 2022
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
Equals 2F1(-1/4,1/4 ; 1/2 ; 3/4) . - R. J. Mathar, Aug 31 2025

Last term corrected and more terms added by Harry J. Smith, Jun 02 2009

A073010 Decimal expansion of Pi/sqrt(27).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

Original name: Decimal expansion of Sum_{n>0} 1/(n*binomial(2*n,n)).
This appears to be Pi/sqrt(27). See A111510. - Marco Matosic, Feb 27 2008
This is Pi*sqrt(3)/9 = A019676*A002194, see eq. (12) in Lehmer link. - R. J. Mathar, Mar 04 2009
Value of the Dirichlet L-series of the non-principal character modulo m=3 (A102283) at s=1. - R. J. Mathar, Oct 03 2011
Construct the largest possible circle inside a given equilateral triangle. This constant is the ratio of the area of the circle to the area of the triangle (A245670 is analogous square in triangle). - Rick L. Shepherd, Jul 29 2014

			0.60459978807807261686469275254738524409468...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (81), page 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005), 25-36.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019. See r(q) for q=3 in Table 1, p. 7.
D. H. Lehmer, Interesting Series Involving the Central Binomial Coefficient, Am. Math. Monthly 92 (1985) 449-457. See eq. (12).
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991), 53-55.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (1.6.2)
Simon Plouffe, Sum(1/(n*binomial(2*n,n)), n=1..infinity), see p. 87.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 501.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
Index entries for transcendental numbers.

Cf. A002194, A019676, A111510, A245670, A248682.

R:=RealField(106); SetDefaultRealField(R); n:=Pi(R)/Sqrt(27); Reverse(Intseq(Floor(10^105*n))); // Bruno Berselli, Mar 12 2018

RealDigits[ N [Sum[1/(n*Binomial[2n, n]), {n, 1, Infinity}], 110]] [[1]]
RealDigits[Pi/(3*Sqrt[3]), 10, 105][[1]] (* T. D. Noe, Sep 11 2013 *)

Pi/sqrt(27) \\ Charles R Greathouse IV, Sep 11 2013

-Pi/(3*sqrt(3)) = Sum_{n>=0} (1/(6*n+1) - 2/(6*n+2) - 3/(6*n+3) - 1/(6*n+4) + 2/(6*n+5) + 3/(6*n+6)). - Mats Granvik, Sep 08 2013
Equals Integral_{0..oo} 2*x/((x^2+1)*(x^4+x^2+1)) dx. - Jean-François Alcover, Sep 10 2013
From Peter Bala, Feb 16 2015: (Start)
Pi/sqrt(27) = Sum_{n >= 0} 1/((3*n + 1)*(3*n + 2)) = 1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ....
Continued fraction: 1/(1 + 1^2/(1 + 2^2/(2 + 4^2/(1 + 5^2/(2 + ... + (3*n + 1)^2/(1 + (3*n + 2)^2/(2 + ... ))))))).
Pi/sqrt(27) = Integral_{t = 0..1/2} 1/(t^2 - t + 1) dt = Integral_{t = 0..1/2} (1 + t - t^3 - t^4)/(1 - t^6) dt.
Pi/sqrt(27) = (1/4)*Sum_{n >= 0} (-1/8)^n * (9*n + 5)/((3*n + 1)*(3*n + 2)).
BBP-type formulas:
Pi/sqrt(27) = Sum_{n >= 0} (1/64)^(n+1)*( 32/(6*n + 1) + 16/(6*n + 2) - 4/(6*n + 4) - 2/(6*n + 5) ) follows from the above integral representation.
Pi/sqrt(27) = Sum_{n >= 0} (-1)^n*(1/27)^(n+1)*( 9/(6*n + 1) + 9/(6*n + 2) + 6/(6*n + 3) + 3/(6*n + 4) + 1/(6*n + 5) ) follows from the result: Pi/3 = Integral_{t = 0..1/sqrt(3)} 1/(1 - sqrt(3)*t + t^2) dt. (End)
Equals Integral_{x=0..oo} x*I_0(x)*K_0(x)^2 dx over a triple product of modified Bessel functions. - R. J. Mathar, Oct 14 2015
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(exp(x) + exp(-x) + 1) dx.
Equals Integral_{x=0..oo} 1/(1 + x + x^2 + x^3 + x^4 + x^5) dx. (End)
Equals (3*S - 4)/8, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Product_{p prime} (1 - Kronecker(-3, p)/p)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p)^(-1). - Amiram Eldar, Nov 06 2023
From Peter Bala, Dec 09 2023: (Start)
Pi/sqrt(27) = Sum_{n >= 1} 1/(n*binomial(2*n,n)) = (1/6)*Sum_{n >= 1} 3^n/(n*binomial(2*n,n)) (see Lehmer, equation 12, and also p. 456).
Pi/sqrt(27) = (1/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n,n)).
Pi/sqrt(27) = (9/4)*Sum_{n >= 3} (n - 1)*(n - 2)/binomial(2*n,n). (End)
Equals integral_{x=0..oo} 1/(1-x^3) dx [Nahin]. - R. J. Mathar, May 16 2024
From Peter Bala, Mar 05 2025: (Start)
Equals 2*Integral_{x = 0..1} 1/(3 + x^2) dx = Integral_{x = 0..1} (4 - x)/(sqrt(x)*(12 + x*(1 - x))) dx.
Equals Sum_{n >= 1} (-1/3)^n * (3 - 14*n)/(n*(2*n-1)*binomial(4*n, 2*n)). The series terms are O(7*sqrt(2*Pi/n)/48^n). (End)
Equals Integral_{x=0..oo} (x^3)/(x^6 + 1) dx. - Kritsada Moomuang, Jun 04 2025

A020760 Decimal expansion of 1/sqrt(3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - Frank M Jackson, Jun 15 2011
When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - Stanislav Sykora, Mar 07 2016
The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - Amiram Eldar, Sep 01 2020
The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - Michal Paulovic, Mar 21 2023

			0.577350269189625764509148780501957455647601751270126876018602326....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.17, pp. 495, 531.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ee Hou Yong and L. Mahadevan, Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, preprint, arXiv:1008.4559 [physics.class-ph], 2010-2011.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (sqrt(3)), A010701 (1/3).

Cf. A002193, A047235, A165952, A195696, A040001.

evalf(1/sqrt(3)); # Michal Paulovic, Mar 21 2023

RealDigits[N[1/Sqrt[3],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

\\ Works in v2.15.0; n = 100 decimal places
my(n=100); digits(floor(10^n/quadgen(12))) \\ Michal Paulovic, Mar 21 2023

Equals 1/A002194. - Michel Marcus, Oct 12 2014
Equals cosine of the magic angle: cos(A195696). - Stanislav Sykora, Mar 07 2016
Equals square root of A010701. - Michel Marcus, Mar 07 2016
Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - Gerry Martens, Nov 22 2022
Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - Gerry Martens, Nov 22 2022
Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - Michal Paulovic, Mar 21 2023
Equals Product_{k>=1} (1 + (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024
Equals tan(Pi/6) = (1/2)/A010527. - R. J. Mathar, Aug 31 2025

A040001 1 followed by {1, 2} repeated.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

Continued fraction for sqrt(3).
Also coefficient of the highest power of q in the expansion of the polynomial nu(n) defined by: nu(0)=1, nu(1)=b and for n>=2, nu(n)=b*nu(n-1)+lambda*(n-1)_q*nu(n-2) with (b,lambda)=(1,1), where (n)_q=(1+q+...+q^(n-1)) and q is a root of unity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
nu(0)=1 nu(1)=1; nu(2)=2; nu(3)=3+q; nu(4)=5+3q+2q^2; nu(5)=8+7q+6q^2+4q^3+q^4; nu(6)=13+15q+16q^2+14q^3+11q^4+5q^5+2q^6.
From Jaroslav Krizek, May 28 2010: (Start)
a(n) = denominators of arithmetic means of the first n positive integers for n >= 1.
See A026741(n+1) or A145051(n) - denominators of arithmetic means of the first n positive integers. (End)
From R. J. Mathar, Feb 16 2011: (Start)
This is a prototype of multiplicative sequences defined by a(p^e)=1 for odd primes p, and a(2^e)=c with some constant c, here c=2. They have Dirichlet generating functions (1+(c-1)/2^s)*zeta(s).
Examples are A153284, A176040 (c=3), A040005 (c=4), A021070, A176260 (c=5), A040011, A176355 (c=6), A176415 (c=7), A040019, A021059 (c=8), A040029 (c=10), A040041 (c=12). (End)
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A000325(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
For n > 0: denominators of row sums of the triangular enumeration of rational numbers A226314(n,k) / A054531(n,k), 1 <= k <= n; see A226555 for numerators. - Reinhard Zumkeller, Jun 10 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)

			1.732050807568877293527446341... = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))
G.f. = 1 + x + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).
Michael Somos, Rational Function Multiplicative Coefficients.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000034, A002194, A133566, A083329 (binomial Transf).

Apart from a(0) the same as A134451.

a040001 0 = 1; a040001 n = 2 - mod n 2
a040001_list = 1 : cycle [1, 2]  -- Reinhard Zumkeller, Apr 16 2015

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[3],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
PadRight[{1},120,{2,1}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = 2 - (n==0) - (n%2)} /* Michael Somos, Jun 11 2003 */

{ allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)); for (n=0, 20000, write("b040001.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

Multiplicative with a(p^e) = 2 if p even; 1 if p odd. - David W. Wilson, Aug 01 2001
G.f.: (1 + x + x^2) / (1 - x^2). E.g.f.: (3*exp(x)-2*exp(0)+exp(-x))/2. - Paul Barry, Apr 27 2003
a(n) = (3-2*0^n +(-1)^n)/2. a(-n)=a(n). a(2n+1)=1, a(2n)=2, n nonzero.
a(n) = sum{k=0..n, F(n-k+1)*(-2+(1+(-1)^k)/2+C(2, k)+0^k)}. - Paul Barry, Jun 22 2007
Row sums of triangle A133566. - Gary W. Adamson, Sep 16 2007
Euler transform of length 3 sequence [ 1, 1, -1]. - Michael Somos, Aug 04 2009
Moebius transform is length 2 sequence [ 1, 1]. - Michael Somos, Aug 04 2009
a(n) = sign(n) + ((n+1) mod 2) = 1 + sign(n) - (n mod 2). - Wesley Ivan Hurt, Dec 13 2013

A010464 Decimal expansion of square root of 6.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 2 followed by {2, 4} repeated. - Harry J. Smith, Jun 05 2009
Ratio t*o/c^2 where t, o and c are respectively the edge lengths of a tetrahedron, an octahedron and a cube whose total surface areas are the same. See CNRS links. - Michel Marcus, Mar 03 2022 and Apr 21 2016
Diameter of a sphere whose surface area equals 6*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Aug 29 2024

			2.449489742783178098197284074705891391965947480656670128432692567250960...
Sqrt(6) = sqrt(1+i*sqrt(3)) + sqrt(1-i*sqrt(3)), where i=sqrt(-1). - _Bruno Berselli_, Nov 20 2012

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b
R. J. Nemiroff and J. Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 6
R. J. Nemiroff and J. Bonnell, The first 1 million digits of the square root of 6
Ana Rechtman, Avril 2016, 2e défi, Images des Mathématiques, CNRS, 2016.
Ana Rechtman, Avril 2016, 3e défi (Solution du 2e défi d’Avril), Images des Mathématiques, CNRS, 2016.
Index entries for algebraic numbers, degree 2

Cf. A002193 (sqrt(2)), A002194 (sqrt(3)).

Cf. A040003 (continued fraction).

SetDefaultRealField(RealField(100)); Sqrt(6); // Vincenzo Librandi, Feb 15 2020

RealDigits[N[Sqrt[6],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

default(realprecision, 20080); x=sqrt(6); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010464.txt", n, " ", d));  \\ Harry J. Smith, Jun 01 2009

PARI
```
sqrt(6) \\ Michel Marcus, Apr 21 2016
```

Equals A002193*A002194. - Michel Marcus, Apr 21 2016

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285

Offset: 1

N. J. A. Sloane, R. K. Guy

The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n))) = 1. - Olivier Gérard Feb 15 1999
From Stanislav Sykora, May 02 2016: (Start)
The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.
The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).
Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024

			34 is a term of this sequence because a circle can be divided into exactly 34 parts. 7 is not.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
Duane W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108. - N. J. A. Sloane, Aug 05 2021
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.

Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^100; terms 1..2000 from T. D. Noe)
Laura Anderson, Jasbir S. Chahal and Jaap Top, The last chapter of the Disquisitiones of Gauss, arXiv:2110.01355 [math.HO], 2021.
Wayne Bishop, How to construct a regular polygon, Amer. Math. Monthly 85(3) (1978), 186-188.
Alessandro Chiodo, A note on the construction of the Śrī Yantra, Sorbonne Université (Paris, France, 2020).
T. Chomette, Construction des polygones réguliers (in French).
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, Amer. Math. Monthly 98(2) (1991), 97-108.
Bruce Director, Measurement and Divisibility.
David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit), Numberphile video (2015).
Mauro Fiorentini, Construibili (numeri).
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 463. Original (in Latin).
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag. 63(1) (1990), 3-20. [Annotated scanned copy] [DOI]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Johann Gustav Hermes, Über die Teilung des Kreises in 65537 gleiche Teile (About the division of the circle into 65537 equal pieces), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Vol. 3 (1894), 170-186.
Friedrich Julius Richelot, De resolutione algebraica aequationis X^257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (On the resolution of the algebraic equation X^257 = 1, or ...), Journal für die reine und angewandte Mathematik 9 (1832), 1-26.
Pierre Wantzel, Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas (Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass), Journal de Mathématiques Pures et Appliquées 2 (1837), 366-372.
Eric Weisstein's World of Mathematics, Constructible Number.
Eric Weisstein's World of Mathematics, Constructible Polygon.
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Trigonometry.
Eric Weisstein's World of Mathematics, Trigonometry Angles.
Wikipedia, Constructible polygon.
Wikipedia, Johann Gustav Hermes.
Wikipedia, Friedrich Julius Richelot.
Wikipedia, Mohr-Mascheroni theorem.
Wikipedia, Pierre Wantzel.
Wikipedia, Poncelet-Steiner theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.

Subsequence of A295298. - Antti Karttunen, Nov 27 2017
A004729 and A051916 are subsequences. - Reinhard Zumkeller, Mar 20 2010
Cf. A000079, A004169, A000215, A099884, A019434 (Fermat primes).
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).
Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).
Cf. also A046528.
Cf. A000079, A092506.

a003401 n = a003401_list !! (n-1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
-- Reinhard Zumkeller, Jul 31 2012

Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (* Robert G. Wilson v, Jun 11 2005 *)
nn=10; logs=Log[2,{2,3,5,17,257,65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i,j,k,l,m,n}.logs; If[z<=nn, Sow[2^z]], {i,0,lim2}, {j,0,1}, {k,0,1}, {l,0,1}, {m,0,1}, {n,0,1}]][[2,1]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)

for(n=1,10^4,my(t=eulerphi(n));if(t/2^valuation(t,2)==1,print1(n,", "))); \\ Joerg Arndt, Jul 29 2014

is(n)=n>>=valuation(n,2); if(n<7, return(n>0)); my(k=logint(logint(n,2),2)); if(k>32, my(p=2^2^k+1); if(n%p, return(0)); n/=p; unknown=1; if(n%p==0, return(0)); p=0; if(is(n)==0, 0, "unknown [has large Fermat number in factorization]"), 4294967295%n==0) \\ Charles R Greathouse IV, Jan 09 2022

is(n)=n>>=valuation(n,2); 4294967295%n==0 \\ valid for n <= 2^2^33, conjecturally valid for all n; Charles R Greathouse IV, Jan 09 2022

from sympy import totient
A003401_list = [n for n in range(1,10**4) if format(totient(n),'b').count('1') == 1]
# Chai Wah Wu, Jan 12 2015

Terms from 3 onward are computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos Elemer, Oct 19 2001, clarified by Antti Karttunen, Nov 27 2017
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1). - Sergio Pimentel, Apr 30 2004, edited by Franklin T. Adams-Watters, Jun 16 2006
If the well-known conjecture that there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4 is true, then we have exactly: Sum_{n>=1} 1/a(n)= 2*Product_{k=0..4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... - Vladimir Shevelev and T. D. Noe, Dec 01 2010
log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015

Definition clarified by Bill Gosper. - N. J. A. Sloane, Jun 14 2020

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.
Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - R. J. Mathar, Jul 26 2007
This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - Jean-François Alcover, Dec 19 2016
Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - Charles R Greathouse IV, May 21 2022
For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022
Surface area of a triangular bipyramid (Johnson solid J_12) with unit edges. - Paolo Xausa, Aug 04 2025

			2.59807621135331594029116951225880855041420788071557094208371046917789952536320...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

G. C. Greubel, Table of n, a(n) for n = 1..10000
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Kiran S. Kedlaya, A < B, (1999), Problem 6.1, p. 6.
J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.
Michael Penn, not as bad as it seems..., YouTube video, 2021.
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Eric Weisstein's World of Mathematics, Twenty-Vertex Entropy Constant.
Wikipedia, Hexagon.
Wikipedia, Regular polygon.
Wikipedia, Triangular bipyramid.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A104954, A104955, A104957.

Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

3*sqrt(3)/2 \\ G. C. Greubel, Jul 03 2017

Equals (3*sqrt(3))/2, that is, 2*A104954.

Equals Product_{k>=3} (((k - 1)^2*(k + 2))/((k + 1)^2*(k - 2)))^(k/2). - Antonio Graciá Llorente, Oct 13 2024

cp's OEIS Frontend

A370944 Decimal expansion of ((1+sqrt(5))/2)*sqrt(3) = A001622*A002194.

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A007310 Numbers congruent to 1 or 5 mod 6.

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A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

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A010527 Decimal expansion of sqrt(3)/2.

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A073010 Decimal expansion of Pi/sqrt(27).

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A020760 Decimal expansion of 1/sqrt(3).

A370944 Decimal expansion of ((1+sqrt(5))/2)sqrt(3) = A001622A002194.