A076214 -id:A076214 - cp's OEIS frontend

A058891 a(n) = 2^(2^(n-1) - 1).

1, 2, 8, 128, 32768, 2147483648, 9223372036854775808, 170141183460469231731687303715884105728, 57896044618658097711785492504343953926634992332820282019728792003956564819968

Offset: 1

N. J. A. Sloane, Jan 08 2001

For n > 1, a(n) is the least solution > 1 to rad(x)^(n-1) = tau(x) where rad(x) = A007947(x) is the squarefree kernel of x and tau(x) = A000005(x) the number of divisors of x. - Benoit Cloitre, Apr 18 2002 [Corrected by Michel Marcus, Oct 15 2018]
For n > 1, a(n) is also the total number of possible outcomes of a knockout tournament starting with 2^(n-1) players, taking account of all matches in the tournament. - Martin Griffiths, Mar 26 2009
Also, a(n+1) = 2^(2^n-1) for n >= 1 are solutions x = y of the Diophantine equation x^y * y^x = (x+y)^z in positive integers; corresponding solutions z are in A348332 (see this last sequence for more informations and links). - Bernard Schott, Oct 13 2021
For n > 2, a(n) ends with 8. - Bernard Schott, Oct 20 2021
a(n) is the number of labeled hypergraphs on n - 1 vertices. - Lorenzo Sauras Altuzarra, Apr 01 2023

			The 8 possible hyperedge sets for the vertex set {1, 2} are {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {2}}, {{1}, {1, 2}}, {{2}, {1, 2}} and {{1}, {2}, {1, 2}}. - _Lorenzo Sauras Altuzarra_, Apr 01 2023

F. Harary, Graph Theory, Page 209, Problem 16.11.

Harry J. Smith, Table of n, a(n) for n = 1..12
Wikipedia, Hypergraph.

Cf. A000079, A000225, A053287, A076214, A077585, A348332.

a[1]:=1: for n from 2 to 20 do a[n]:=2*a[n-1]^2 od: seq(a[n], n=1..9); # Zerinvary Lajos, Apr 16 2009

a = 1; b = -3; Table[Expand[(-1/2) ((a + Sqrt[b])^(2^n) + (a - Sqrt[b])^(2^n))], {n, 1, 10}] (* Artur Jasinski, Oct 11 2008 *)

a(n) = { 2^(2^(n-1)-1) } \\ Harry J. Smith, Jun 23 2009

def A058891(n): return 1<<(1<Chai Wah Wu, Dec 12 2022

a(n) = A053287(A000079(n-1)).
a(1) = 1, a(n+1) = 2*a(n)^2.
a(1) = 1, a(n+1) = 2^n*a(1)*a(2)*...*a(n). - Benoit Cloitre, Sep 13 2003
a(n) = (-1/2)*((1 + sqrt(-3))^(2^n) + (1 - sqrt(-3))^(2^n)). - Artur Jasinski, Oct 11 2008
a(n) = 2*a(n-1)^2 is an example with a(1) = 1 and k = 2 of a(n) = k*a(n-1)^2; general explicit formula: a(n) = ((a(1)*k)^(2^(n-1)))/k. - Andreas Pfaffel (andreas.pfaffel(AT)gmx.at), Apr 27 2010
a(n) = A077585(n-1) + 1. - Maurizio De Leo, Feb 25 2015
a(n) = 2^A000225(n-1). - Michel Marcus, Aug 19 2020
Sum_{n>=0} 1/a(n) = A076214. - Amiram Eldar, Oct 27 2020

A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Labos Elemer

Numbers k such that tau(k)^phi(k) == 1 (mod k), where tau(k) is the number of divisors of k (A000005) and phi(k) is the Euler phi function (A000010). - Michel Lagneau, Nov 20 2012
Density is at least 4/Pi^2 = 0.405... since A056911 is a subsequence, and at most 1/2 since all even numbers in this sequence are squares. The true value seems to be around 0.4504. - Charles R Greathouse IV, Mar 27 2013
They are called anti-tau numbers by Zelinsky (see link) and their density is at least 3/Pi^2 (theorem 57 page 15). - Michel Marcus, May 31 2015
From Amiram Eldar, Feb 21 2021: (Start)
Spiro (1981) proved that the number of terms of this sequence that do not exceed x is c * x + O(sqrt(x)*log(x)^3), where 0 < c < 1 is the asymptotic density of this sequence.
The odd numbers whose number of divisors is a power of 2 (the odd terms of A036537) are terms of this sequence. Their asymptotic density is A327839/A076214 = 0.4212451116... which is a better lower bound than 4/Pi^2 for the asymptotic density of this sequence.
A better upper limit than 0.5 can be obtained by considering the subsequence of odd numbers whose 3-adic valuation is not of the form 3*k-1 (i.e., odd numbers without those k with gcd(k, tau(k)) = 3), whose asymptotic density is 6/13 = 0.46153...
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 49, 459, 4535, 45145, 450710, 4504999, 45043234, 450411577, 4504050401, ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mart Abel, Helena Lauer, and Ellen Redi, About the number of τ-numbers relative to polynomials with integer coefficients, Acta Comment. Univ. Tartu. Math. 25, No. 1, 107-117, 2021.
Claudia A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 1981.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A000010, A009191, A009230, A036537, A056911, A076214, A327839.

a046642 n = a046642_list !! (n-1)
a046642_list = map (+ 1) $ elemIndices 1 a009191_list
-- Reinhard Zumkeller, Aug 14 2011

Select[ Range[200], CoprimeQ[#, DivisorSigma[0, #]] &] (* Jean-François Alcover, Oct 20 2011 *)

is(n)=gcd(numdiv(n),n)==1 \\ Charles R Greathouse IV, Mar 27 2013

A009191(a(n)) = 1.

A006466 Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

C arises when looking for a sequence b(n) such that b(1)=0 and b(n+1) is the smallest integer > b(n) such that the continued fraction for 1/2^b(1) + 1/2^b(2) + ... + 1/2^b(n+1) contains only 1's or 2's. It arises because b(n) = 2^n - 1 and C = Sum_{k>=0} 1/2^b(k). - Benoit Cloitre, Nov 03 2002

			1.632843018043786287416159475... = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 09 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), #13.2.15.
J. Shallit, Letter to N. J. A. Sloane with attachment, Aug. 1979
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217 [DOI]

Cf. A076214 = Decimal expansion. - Harry J. Smith, May 09 2009

{ allocatemem(932245000); default(realprecision, 10000); x=suminf(n=0, 1/2^(2^n)); x=contfrac(2*x); for (n=1, 20001, write("b006466.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 09 2009

Recurrence: a(5n) = a(5n+1) = a(2) = a(5n+3) = a(20n+14) = a(40n+9) = 1, a(20n+4) = a(40n+29) = 2, a(5n+2) = 3 - a(5n-1), a(20n+19) = a(10n+9). - Ralf Stephan, May 17 2005

Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001

A283526 Pierce expansion of the number Sum_{k >= 1} 1/(2^(2^k - 1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 16, 17, 256, 257, 65536, 65537, 4294967296, 4294967297, 18446744073709551616, 18446744073709551617, 340282366920938463463374607431768211456, 340282366920938463463374607431768211457

Offset: 0

Kutlwano Loeto, Mar 10 2017

nonn

This sequence is the Pierce expansion of the number 2*s(2) - 1 = 0.632843018043786287416159475061... where s(u) = Sum_{k>=0} 1/u^(2^k) that has been considered by J. Shallit in A007400. The continued fraction expansion of this number is essentially A006466.

			The Pierce expansion of 0.6328430180437862 starts as 1 - 1/2 + 1/(2*3) - 1/(2*3*4) + 1/(2*3*4*5) - 1/(2*3*4*5*16) + ...

Michael De Vlieger, Table of n, a(n) for n = 0..24
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217.

Cf. A006466, A007400, A076214.

L:=[1]: for k from 0 to 6 do: L:=[op(L),2^(2^k),2^(2^k)+1]: od: print(L);

{1}~Join~Map[{#, # + 1} &, 2^2^Range[0, 8]] // Flatten (* Michael De Vlieger, Mar 18 2017 *)

a(0) = 1, a(2k+1) = 2^(2^k), a(2k+2) = 2^(2^k) + 1.

A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			0.50781250046566128730773925781250000000587747175411...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346191, A346192.

RealDigits[Sum[1/(2^(2^(2*n + 1) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346191.

A346191 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k)) - 1).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			1.12503051757812500010842021724855044340074528008699...

Wolfgang Schwarz, Remarks on the irrationality and transcendence of certain series, Mathematica Scandinavica, Vol. 20, No. 2 (1967), pp. 269-274; alternative link.
Index entries for transcendental numbers

Cf. A048649, A076214, A346190, A346192.

RealDigits[Sum[1/(2^(2^(2*n) - 1)), {n, 0, 10}], 10, 100][[1]]

Equals A076214 - A346190.

cp's OEIS Frontend

A058891 a(n) = 2^(2^(n-1) - 1).

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A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

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A006466 Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n).

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A283526 Pierce expansion of the number Sum_{k >= 1} 1/(2^(2^k - 1)).

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A346190 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k+1)) - 1).

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A346191 Decimal expansion of Sum_{k>=0} 1/(2^(2^(2*k)) - 1).

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