A003401 -id:A003401 - cp's OEIS frontend

A113402 Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).

1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 512

Offset: 1

Eric W. Weisstein, Oct 28 2005

a(n) is always a power of 2.
It would appear that a(n) <= a(n+1) and that for a(n)=2^k, the count for k beginning with 0 is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, ...; or that the count for k is k+2 for k > 0. - Robert G. Wilson v, Jul 31 2014
Apparently v_2(a(n)) = A052146(n-1) for n >= 2 where v_2 is the 2-adic valuation. - Joerg Arndt, Jul 29 2014 [incorrect for n >= 561, Joerg Arndt, Mar 03 2019]

Robert G. Wilson v, Table of n, a(n) for n = 1..1632
Eric Weisstein's World of Mathematics, Trigonometry Angles

Cf. A000217, A002024, A003401, A055034, A113401.

f[n_] := Exponent[MinimalPolynomial[Cos[Pi/n]][x], x]; Table[ f@ n, {n, Select[ Range@ 1300, IntegerQ[ Log[2, EulerPhi[#]]] &]}] (* Robert G. Wilson v, Jul 28 2014 *)
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 2500; t = Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}]; f[n_] := EulerPhi[ 2n]/2; f[1] = 1; f@# & /@ t (* Robert G. Wilson v, Jul 28 2014 *)

A094269 a(n) = phi(A003401(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 4, 4, 4, 8, 8, 16, 8, 8, 8, 16, 16, 16, 16, 32, 16, 32, 32, 32, 64, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 128, 128, 256, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 512, 256, 256, 256, 512, 512, 512, 512, 1024, 512, 512, 512, 512

Offset: 1

Labos Elemer, Jun 02 2004

nonn

All terms are powers of 2. - Jianing Song, Sep 28 2018

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000010, A003401, A319821.

Do[If[IntegerQ[Log[2, EulerPhi[n]]], Print[n];ta[[u]]=n;u=u+1], {n, 1, 10000}] EulerPhi[ta]

for(n=1, 1000, my(i=eulerphi(n)); if(omega(2*i)==1, print1(i, “, “))) \\ Jianing Song, Sep 28 2018

a(n) = 2^A319821(n). - Amiram Eldar, Sep 25 2024

A113401 Algebraic degree of cos(2*Pi/n) for constructible n-gons (A003401).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, 8, 8, 8, 16, 8, 16, 16, 16, 32, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 128, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 256, 128, 128, 128, 256, 256, 256, 256, 512, 256, 256, 256, 256, 256, 256, 512

Offset: 1

Eric W. Weisstein, Oct 28 2005

Eric Weisstein's World of Mathematics, Trigonometry Angles

Cf. A003401, A023022, A113402.

A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 1, 2, 2, 2, 0, 1, 2, 3, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 0, 3, 1, 3, 2, 3, 3, 3, 1, 2, 3, 4, 2, 3, 3, 4, 1, 4, 2, 2, 2, 3, 3, 3, 2, 4, 3, 4, 2, 3, 3, 4, 0, 3, 3, 4, 1, 4, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 1, 4, 2, 3, 3, 2, 4, 4, 2, 3, 3, 4, 3, 4, 4, 4, 1, 2, 4, 4, 2

Offset: 1

Ali Sada and Robert G. Wilson v, Feb 28 2020

From Antti Karttunen, Apr 07 2020: (Start)
Also the least number of iterations of nondeterministic map k -> k-(k/p) needed to reach a power of 2, when any prime factor p of k can be used. The minimal length path to the nearest power of 2 (= 2^A064415(n)) is realized whenever one uses any of the A005087(k) distinct odd prime factors of the current k, at any step of the process. For example, this could be done by iterating with the map k -> k-(k/A078701(k)), i.e., by using the least odd prime factor of k (instead of the largest prime).
Proof: Viewing the prime factorization of changing k as a multiset ("bag") of primes, we see that liquefying any odd prime p with step p -> (p-1) brings at least one more 2 to the bag, while applying p -> (p-1) to any 2 just removes it from the bag, but gives nothing back. Thus the largest (and thus also the nearest) power of 2 is reached by eliminating - step by step - all odd primes from the bag, but none of 2's, and it doesn't matter in which order this is done.
The above implies also that the sequence is totally additive, which also follows because both A064097 and A064415 are. That A064097(n) = A329697(n) + A054725(n) for all n > 1 can be also seen by comparing the initial conditions and the recursion formulas of these three sequences.
For any n, A333787(n) is either the nearest power of 2 reached (= 2^A064415(n)), or occurs on some of the paths from n to there.
(End)
A003401 gives the numbers k where a(k) = A005087(k). See also A336477. - Antti Karttunen, Mar 16 2021

			The trajectory of 15 is {12, 8}, taking 2 iterations to reach 8 = 2^3. So a(15) is 2.
From _Antti Karttunen_, Apr 07 2020: (Start)
Considering all possible paths from 15 to 1 nondeterministic map k -> k-(k/p), where p can be any prime factor of k, we obtain the following graph:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \__  |  __/|
      \_|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.
It can be seen that there's also alternative route to 8 via 10 (with 10 = 15-(15/3), where 3 is not the largest prime factor of 15), but it's not any shorter than the route via 12.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Michael De Vlieger, Annotated fan style binary tree labeling the index n, with a color code where black represents a(n) = 0, red a(n) = 1, and magenta the largest value in a(n) for n = 1..16383.

Cf. A000265, A003401, A005087, A052126, A053575, A054725, A064097, A064415, A078701, A087436, A147545, A171462, A209229, A333123, A333787, A333790, A334107, A334109, A335875, A334204, A335880, A335881, A336396, A336466, A336469 [= a(phi(n))], A336928 [= a(sigma(n))], A336470, A336477, A339970.
Cf. A000079, A334101, A334102, A334103, A334104, A334105, A334106 for positions of 0 .. 6 in this sequence, and also array A334100.
Cf. A334099 (a right inverse, positions of the first occurrence of each n).
Cf. A334091 (first differences), A335429 (partial sums).
Cf. also A331410 (analogous sequence when using the map k -> k + k/p), A334861, A335877 (their sums and differences), see also A335878 and A335884, A335885.

a[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, n, # != 2^IntegerExponent[#, 2] &] -1; Array[a, 100]

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1])))); \\ Antti Karttunen, Apr 07 2020

up_to = 2^24;
A329697list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = if(!bitand(n,n-1),0,1+vecmin(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };
v329697 = A329697list(up_to);
A329697(n) = v329697[n]; \\ Antti Karttunen, Apr 07 2020

A329697(n) = if(n<=2,0, if(isprime(n), A329697(n-1)+1, my(f=factor(n)); (apply(A329697, f[, 1])~ * f[, 2]))); \\ Antti Karttunen, Apr 19 2020

From Antti Karttunen, Apr 07-19 2020: (Start)
a(1) = a(2) = 0; and for n > 2, a(p) = 1 + a(p-1) if p is an odd prime and a(n*m) = a(n) + a(m) if m,n > 1. [This is otherwise equal to the definition of A064097, except here we have a different initial condition, with a(2) = 0].
a(2n) = a(A000265(n)) = a(n).
a(p) = 1+a(p-1), for all odd primes p.
If A209229(n) == 1 [when n is a power of 2], a(n) = 0,
  otherwise a(n) = 1 + a(n-A052126(n)) = 1 + a(A171462(n)).
Equivalently, for non-powers of 2, a(n) = 1 + a(n-(n/A078701(n))),
or equivalently, for non-powers of 2, a(n) = 1 + Min a(n - n/p), for p prime and dividing n.
a(n) = A064097(n) - A064415(n), or equally, a(n) = A064097(n) - A054725(n), for n > 1.
a(A019434(n)) = 1, a(A334092(n)) = 2, a(A334093(n)) = 3, etc. for all applicable n.
For all n >= 0, a(A334099(n)) = a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = a(257^n) = a(65537^n) = n.
a(A122111(n)) = A334107(n), a(A225546(n)) = A334109(n).
(End)
From Antti Karttunen, Mar 16 2021: (Start)
a(n) = a(A336466(n)) + A087436(n) = A336396(n) + A087436(n).
a(A053575(n)) = A336469(n) = a(n) - A005087(n).
a(A147545(n)) = A000120(A147545(n)) - 1.
(End)

A092506 Prime numbers of the form 2^n + 1.

Original entry on oeis.org

2, 3, 5, 17, 257, 65537

Offset: 1

Jorge Coveiro, Apr 05 2004

2 together with the Fermat primes A019434.
Obviously if 2^n + 1 is a prime then either n = 0 or n is a power of 2. - N. J. A. Sloane, Apr 07 2004
Numbers m > 1 such that 2^(m-2) divides (m-1)! and m divides (m-1)! + 1. - Thomas Ordowski, Nov 25 2014
From Jaroslav Krizek, Mar 06 2016: (Start)
Also primes p such that sigma(p-1) = 2p - 3.
Also primes of the form 2^n + 3*(-1)^n - 2 for n >= 0 because for odd n, 2^n - 5 is divisible by 3.
Also primes of the form 2^n + 6*(-1)^n - 5 for n >= 0 because for odd n, 2^n - 11 is divisible by 3.
Also primes of the form 2^n + 15*(-1)^n - 14 for n >= 0 because for odd n, 2^n - 29 is divisible by 3. (End)
Exactly the set of primes p such that any number congruent to a primitive root (mod p) must have at least one prime divisor that is also congruent to a primitive root (mod p). See the links for a proof. - Rafay A. Ashary, Oct 13 2016
Conjecture: these are the only solutions to the equation A000010(x)+A000010(x-1)=floor((3x-2)/2). - Benoit Cloitre, Mar 02 2018
For n > 1, if 2^n + 1 divides 3^(2^(n-1)) + 1, then 2^n + 1 is a prime. - Jinyuan Wang, Oct 13 2018
The prime numbers occurring in A003401. Also, the prime numbers dividing at least one term of A003401. - Jeppe Stig Nielsen, Jul 24 2019

Rafay A. Ashary, A Property of A092506
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Fermat Number

A019434 is the main entry for these numbers.

Cf. A000010, A000040, A000051, A000215, A003401.

Filtered(List([1..20],n->2^n+1),IsPrime); # Muniru A Asiru, Oct 25 2018

[2^n + 1 : n in [0..25] | IsPrime(2^n+1)]; // Vincenzo Librandi, Oct 14 2018

Select[2^Range[0,100]+1,PrimeQ] (* Harvey P. Dale, Aug 02 2015 *)

print1(2); for(n=0,9, if(ispseudoprime(t=2^2^n+1), print1(", "t))) \\ Charles R Greathouse IV, Aug 29 2016

A099884 XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 15, 16, 24, 20, 30, 17, 32, 48, 40, 60, 34, 51, 64, 96, 80, 120, 68, 102, 85, 128, 192, 160, 240, 136, 204, 170, 255, 256, 384, 320, 480, 272, 408, 340, 510, 257, 512, 768, 640, 960, 544, 816, 680, 1020, 514, 771, 1024, 1536, 1280, 1920

Offset: 0

Paul D. Hanna, Oct 28 2004

Define an "XOR difference triangle" for a sequence A by the following process. Start with A in the leftmost column. Generate the next column by performing the XOR operation between adjacent terms of the prior column. Repeat this process to generate the XOR difference triangle for A. Further, we define the "XOR BINOMIAL transform" of A as the main diagonal in the XOR difference triangle for A. The XOR BINOMIAL transform is its self-inverse. Let a sequence B be the XOR BINOMIAL transform of A, then we may express B by: B(n) = SumXOR_{k=0..n} A047999(n,k)*A(k), which is equivalent to: B(n) = (C(n,0)mod 2)*A(0) XOR (C(n,1)mod 2)*A(1) XOR (C(n,2)mod 2)*A(2) XOR ... XOR (X(n,n)mod 2)*A(n), where the coefficients are C(n,k)(mod 2) = A047999(n,k).

This sequence is a rearrangement of the numbers which are 2^k times distinct Fermat numbers (numbers of the form 2^(2^m) + 1). This matches the sizes of polygons constructible with compass and straightedge (A003401) up to 2^32+1, which is the first nonprime Fermat number. - Franklin T. Adams-Watters, Jun 16 2006

			The main diagonal equals A001317 (Pascal's triangle mod 2 in decimal):
{1,3,5,15,17,51,85,255,257,771,1285,3855,...}, and defines the XOR BINOMIAL transform of the powers of 2.
Rows begin:
  1;
  2, 3;
  4, 6, 5;
  8, 12, 10, 15;
  16, 24, 20, 30, 17;
  32, 48, 40, 60, 34, 51;
  64, 96, 80, 120, 68, 102, 85;
  128, 192, 160, 240, 136, 204, 170, 255;
  256, 384, 320, 480, 272, 408, 340, 510, 257;
  512, 768, 640, 960, 544, 816, 680, 1020, 514, 771;
  1024, 1536, 1280, 1920, 1088, 1632, 1360, 2040, 1028, 1542, 1285;
  2048, 3072, 2560, 3840, 2176, 3264, 2720, 4080, 2056, 3084, 2570, 3855;
  ...
From _Antti Karttunen_, Sep 19 2016: (Start)
Viewed as a square array, the top left corner looks like this:
     1,    2,     4,     8,    16,     32,     64,    128
     3,    6,    12,    24,    48,     96,    192,    384
     5,   10,    20,    40,    80,    160,    320,    640
    15,   30,    60,   120,   240,    480,    960,   1920
    17,   34,    68,   136,   272,    544,   1088,   2176
    51,  102,   204,   408,   816,   1632,   3264,   6528
    85,  170,   340,   680,  1360,   2720,   5440,  10880
   255,  510,  1020,  2040,  4080,   8160,  16320,  32640
   257,  514,  1028,  2056,  4112,   8224,  16448,  32896
   771, 1542,  3084,  6168, 12336,  24672,  49344,  98688
  1285, 2570,  5140, 10280, 20560,  41120,  82240, 164480
  3855, 7710, 15420, 30840, 61680, 123360, 246720, 493440
  4369, 8738, 17476, 34952, 69904, 139808, 279616, 559232
  ...
(End)
The square array shown above can be viewed as a subtable of a multiplication table with particular relevance to the carryless multiplication defined by A048720, as the first column gives the A048720 powers of 3 (and the first row gives powers of 2, which are the same as in standard arithmetic). - _Peter Munn_, Jan 13 2020

Paul D. Hanna, First 45 Rows of Triangle, in flattened form.
SeqFan-mailing list, Discussion of about related array A255483
Index entries for sequences related to polynomials in ring GF(2)[X]

Essentially GF(2)[X] analog of table A036561. - Antti Karttunen, Jan 18 2020
Cf. A047999, A158875 (row sums).
Cf. A000215, A003401, A048675, A048720, A048724, A066117, A248663, A255483, A276586.
Cf. A000079 (first column of triangular table, the topmost row of square array).
Cf. A001317 (the rightmost diagonal of triangular table, the leftmost column of square array).
Cf. A099885, A117998 (central diagonals).
Cf. A276618 (transpose), A091202, A193231.

a[n_]:= Sum[Mod[Binomial[n, i], 2]*2^i, {i, 0, n}]; T[n_, k_]:=2^(n - k)a[k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 11 2017 *)

{T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*2^(n-i)));B}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

from sympy import binomial
def a(n):
    return sum((binomial(n, i)%2)*2**i for i in range(n + 1))
def T(n, k): return 2**(n - k)*a(k)
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 11 2017

(define (A099884 n) (A099884bi (A002262 n) (A025581 n)))
;; Then use either this recurrence:
(define (A099884bi row col) (if (zero? row) (A000079 col) (A048724 (A099884bi (- row 1) col))))
;; or this one:
(define (A099884bi row col) (if (zero? col) (A001317 row) (* 2 (A099884bi row (- col 1)))))
;; Antti Karttunen, Sep 19 2016

T(n, k) = 2^(n-k)*A001317(k). T(n, n) = A001317(n) = SumXOR_{k=0..n} A047999(n, k)*2^k, where SumXOR is the analog of summation under the binary XOR operation.
From Antti Karttunen, Sep 19 2016: (Start)
When viewed as a square array A(row,col), with row >= 0, col >= 0, the following recurrences and formulas are valid:
A(0,col) = A000079(col), for row > 0, A(row,col) = A048724(A(row-1, col)).
A(row,0) = A001317(row), for col > 0, A(row,col) = 2*A(row,col-1).
A(row,col) = A248663(A066117(row+1,col+1)) = A048675(A255483(row,col+1)).
(End)
With the definitions from Antti Karttunen above, A(row+1, col) = A048720(3, A(row, col)). - Peter Munn, Jan 13 2020
A(n,k) = A193231(A(k,n)) = A091202(A036561(n,k)). - Antti Karttunen, Jan 18 2020

Square array interpretation added as a second, alternative description by Antti Karttunen, Sep 19 2016

A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 1, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 1, 5, 1, 3, 3, 9, 9, 3, 1, 5, 3, 21, 5, 3, 11, 23, 1, 21, 5, 1, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 1, 3, 5, 33, 1, 11, 3, 35, 3, 9, 9, 5, 9, 15, 3, 39, 1, 27, 5, 41, 3, 1, 21, 7, 5, 11, 3, 9, 11, 15, 23

Offset: 1

Labos Elemer, Jan 18 2000

This is not necessarily the squarefree kernel. E.g., for n=19, phi(19)=18 is divisible by 9, an odd square. Values at which this kernel is 1 correspond to A003401 (polygons constructible with ruler and compass).

Multiplicative with a(2^e) = 1, a(p^e) = p^(e-1)*A000265(p-1). - Christian G. Bower, May 16 2005

			n = 70 = 2*5*7, phi(70) = 24 = 8*3, so the odd kernel of phi(70) is a(70)=3. [corrected by _Bob Selcoe_, Aug 22 2017]
From _Bob Selcoe_, Aug 22 2017: (Start)
a(89) = 88/8 = 11.
For n = 8820, 8820 = 2^2*3^2*5*7^2; S = 3*5*7 = 105, n" = 3^2*5*7^2 = 2205. a(3)*a(5)*a(7) = 1*1*3 = 3; a(8820) = 3*2205/105 = 63.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A000265, A227944, A329697, A336466, A336468, A336469, A339879, A339971, A339974.

a053575 = a000265 . a000010  -- Reinhard Zumkeller, Oct 09 2013

a:= n-> (t-> t/2^padic[ordp](t, 2))(numtheory[phi](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 14 2020

Array[NestWhile[Ceiling[#/2] &, EulerPhi@ #, EvenQ] &, 94] (* Michael De Vlieger, Aug 22 2017 *) (* or *)
t=Array[EulerPhi,94]; t/2^IntegerExponent[t,2] (* Giovanni Resta, Aug 23 2017 *)

a(n)=n=eulerphi(n);n>>valuation(n,2) \\ Charles R Greathouse IV, Mar 05 2013

From Bob Selcoe, Aug 22 2017: (Start)
Let n" be the odd part of n, S be the odd squarefree kernel of n and p_i {i = 1..z} be all the prime factors of S. Then the sequence can be constructed by the following:
a(1) = 1;
a(n) = (n-1)" when n is prime; and
a(n) = Product_{i = 1..z} a(p_i)*n"/S when n is composite (see Examples).
(End)
From Antti Karttunen, Dec 27 2020: (Start)
a(n) = A336466(n) for squarefree n (see A005117).
A336466(a(n)) = A336468(n), A329697(a(n)) = A336469(n) = A329697(n) - A005087(n).
(End)

A182007 Decimal expansion of 2*sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 06 2012

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

			1.1755705045849462583374119...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Equals sqrt(3-phi).
Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013
Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024
Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024

A053576 Smallest number whose Euler totient is divisible by 2^n.

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776

Offset: 0

Labos Elemer, Jan 18 2000

nonn

n = 32 is the first place where this differs from A001317, since 2^32 + 1 is not prime. - Mitch Harris, May 02 2007
a(8589934592) is the first unknown term; it is 2^8589934593 if F(33) = 2^(2^33)+1 is composite or F(33) otherwise. - Charles R Greathouse IV, Jul 15 2013
a(n) is the only odd element of the set phi-1(2^n), the totient inverses of 2^n. All other elements are 2*a(n), and the even elements of phi-1(2^(n-1)) * 2. - Torlach Rush, Sep 05 2017

			1,2,4,8,...,131072 divide phi of 2,3,5,15,...,196611 = 3*65537 respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 0..3320

Cf. A000010, A003401, A001317, A045544, A058213, A058214, A058215.

Not the same as A001317.

With[{s = Array[EulerPhi, 10^6]}, Table[FirstPosition[s, ?(Divisible[#, 2^n] &)][[1]], {n, 0, 19}]] (* _Michael De Vlieger, Sep 05 2017 *)

a(n)={
  if(n >= 8589934592 && valuation(n>>5,2)>27,
    warning("Result is conjectural on the nonexistence of Fermat primes >= F(33).")
  );
  if(n>31,
    return(2<Charles R Greathouse IV, Jul 15 2013

More odd terms from Jud McCranie, Jan 25 2000

A004729 Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass).

Original entry on oeis.org

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295

Offset: 0

N. J. A. Sloane

The 32 divisors of the product of the 5 known Fermat primes.
The only known odd numbers whose totient is a power of 2. - Labos Elemer, Dec 06 2000
Equals first 32 members of A001317. Also, equals first 32 members of A053576. - Omar E. Pol, Dec 10 2008
Omitting the first term a(0)=1 gives A045544 (the number of sides of constructible odd-sided regular polygons.)

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996; see p. 140.

Eric Weisstein's World of Mathematics, Regular Polygon, Sierpiński Sieve, Constructible Polygon
OEIS Wiki, Constructible odd-sided polygons
OEIS Wiki, Sierpinski's triangle
Index entries for sequences related to divisors of numbers

Essentially same as A045544.

Cf. A000010, A000215, A001317, A003401, A003527, A004169, A004729, A019434, A045544, A047999, A053576, A054432.

Mathematica
```
Divisors[2^32-1]
```
PARI
```
divisors(1<<32-1)
```

Edited by Daniel Forgues, Jun 17 2011

cp's OEIS Frontend

A113402 Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).

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A094269 a(n) = phi(A003401(n)).

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A113401 Algebraic degree of cos(2*Pi/n) for constructible n-gons (A003401).

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A329697 a(n) is the number of iterations needed to reach a power of 2 starting at n and using the map k -> k-(k/p), where p is the largest prime factor of k.

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A092506 Prime numbers of the form 2^n + 1.

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A099884 XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals.

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A053575 Odd part of phi(n): a(n) = A000265(A000010(n)).

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