A087046 -id:A087046 - cp's OEIS frontend

A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

1, 1, 2, 2, 22, 40, 114, 12, 480, 944, 2026, 3918, 8166, 16104, 32630, 240, 131038, 260928, 524250, 1046418, 2096706, 4190168, 8388562, 16768200, 33554240, 67092432, 134216136, 268402446, 536870854, 1073672968, 2147483586, 65280, 8589928346, 17179606976, 34359737478

Offset: 1

Eric W. Weisstein, Apr 28 2006

nonn

a(2^n) is A087046(n).

			The onsets begin at 1, 3, 1+2*sqrt(2), 1+sqrt(6), ...

Cheng Zhang, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Logistic Map

Cf. A000740, A006876, A086178, A086180, A087046, A091517, A098587, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911.

degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; degRo[n_] := degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}]; Table[If[n <= 2, 1, 2 If[2^Round[Log2[n]] == n, degRp[n/2], degRo[n]]], {n, 1, 35}] (* Cheng Zhang, Apr 02 2012 *)

More terms from Cheng Zhang, Apr 02 2012

A220294 a(n) = 1 - 2^(2^n) + 2^(2^(n+1)).

Original entry on oeis.org

3, 13, 241, 65281, 4294901761, 18446744069414584321, 340282366920938463444927863358058659841, 115792089237316195423570985008687907852929702298719625575994209400481361428481

Offset: 0

Michael Somos, Dec 10 2012

nonn

An infinite coprime sequence defined by recursion.

Cf. A001146, A002061, A002716, A087046, A111403, A220161, A255770, A255771, A255772.

[1 - 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018

Table[4^(2^m) - 2^(2^m) + 1, {m, 0, 7}] (* Michael De Vlieger, Aug 02 2016 *)

A220294(n):=1 - 2^(2^n) + 2^(2^(n+1))$ makelist(A220294(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

{a(n) = if( n<0, 0, 1 - 2^(2^n) + 2^(2^(n+1)))};

A220161(n+1) = a(n) * A220161(n).
a(n+1) = 1 + (a(n) - 1) * (A220161(n) - 1).
a(n) = A002716(2*n) = 1 + A087046(n+2) = 1 + A111403(n).
a(n) = A002061(A001146(n)). - Pontus von Brömssen, Aug 31 2021

A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

Original entry on oeis.org

7, 21, 273, 65793, 4295032833, 18446744078004518913, 340282366920938463481821351505477763073, 115792089237316195423570985008687907853610267032561502502920958615344897851393

Offset: 0

Michel Marcus, Dec 06 2012

nonn

For n >= 1, W. Sierpiński proves that a(n) is divisible by 21.

For n >= 1, A. Engel shows that a(n) = a(n-1) * A220294(n-1). - Hans Havermann, Mar 07 2015

Arthur Engel, Problem-Solving Strategies, Springer, 1998, pages 121-122 (E3, said to be a "recent competition problem from the former USSR").
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970. Problem #123.

G. C. Greubel, Table of n, a(n) for n = 0..10
K. Zelator, On a theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers, arXiv:0806.1514 [math.GM], 2008.

Cf. A000215, A002061, A070969, A087046, A220294, A255770, A255771, A255772.

[1 + 2^(2^n) + 2^(2^(n+1)): n in [0..10]]; // G. C. Greubel, Aug 10 2018

Table[1+2^(2^n)+2^(2^(n+1)),{n,0,7}] (* Harvey P. Dale, Dec 16 2015 *)

A220161(n):=1 + 2^(2^n) + 2^(2^(n+1))$ makelist(A220161(n),n,0,10); /* Martin Ettl, Dec 10 2012 */

vector(10, n, n--; 1 + 2^(2^n) + 2^(2^(n+1))) \\ G. C. Greubel, Aug 10 2018

def a(n): return 1 + 2**(2**n) + 2**(2**(n+1))
print([a(n) for n in range(8)]) # Michael S. Branicky, Jul 21 2021

a(n) = A000215(n+1) + A000215(n) - 1.
A070969(n) = sqrt(4*a(n) - 3). a(n+1) = a(n) * (1 + a(n) - A070969(n)) = a(n) * (1 + A087046(n+2)) hence a(n) divides a(n+1). - Michael Somos, Dec 10 2012
a(n) = A002061(A000215(n)). - Pontus von Brömssen, Aug 31 2021

A111403 a(n) = f(f(n+1)) - f(f(n)), where f(m) = 2^m.

Original entry on oeis.org

2, 12, 240, 65280, 4294901760, 18446744069414584320, 340282366920938463444927863358058659840, 115792089237316195423570985008687907852929702298719625575994209400481361428480

Offset: 0

N. J. A. Sloane, Nov 12 2005

nonn

			The binary representation of the first values shows what is going on:
10
1100
11110000
1111111100000000
11111111111111110000000000000000
...

Alois P. Heinz, Table of n, a(n) for n = 0..10

Probably equal to A087046(n+2). Doubled A040996.

a:= n-> (p-> p*(p-1))(2^(2^n)):
seq(a(n), n=0..7);  # Alois P. Heinz, Jan 03 2018

Conjecture: a(n) = A002716(2*n)-1. - R. J. Mathar, May 15 2007
From Alois P. Heinz, Jan 03 2018: (Start)
a(n) = 2^(2*2^n) - 2^(2^n).
a(n) = p*(p-1) with p = 2^(2^n). (End)
a(n) = A040996(n) * 2. - Tilman Piesk, Oct 04 2024

Example and cross-reference from Olivier Gérard, Jun 23 2014

A211667 Number of iterations sqrt(sqrt(sqrt(...(n)...))) such that the result is < 2.

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Apr 30 2012

Different from A001069, but equal for n < 256.

			a(n) = 1, 2, 3, 4, 5, ... for n = 2^1, 2^2, 2^4, 2^8, 2^16, ..., i.e., n = 2, 4, 16, 256, 65536, ... = A001146.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001069, A001146, A010096, A211662, A211668, A211670.

Cf. A087046 (run lengths).

a[n_] := Length[NestWhileList[Sqrt, n, # >= 2 &]] - 1; Array[a, 100] (* Amiram Eldar, Dec 08 2018 *)

apply( A211667(n, c=0)={while(n>=2, n=sqrtint(n); c++); c}, [1..50]) \\ This defines the function A211667. The apply(...) provides a check and illustration. - M. F. Hasler, Dec 07 2018

a(n) = if(n<=1,0, logint(logint(n,2),2) + 1); \\ Kevin Ryde, Jan 18 2024

A211667=lambda n: n and (n.bit_length()-1).bit_length() # Natalia L. Skirrow, May 16 2023

a(2^(2^n)) = a(2^(2^(n-1))) + 1, for n >= 1.
G.f.: g(x) = 1/(1-x)*Sum_{k>=0} x^(2^(2^k))
  = (x^2 + x^4 + x^16 + x^256 + x^65536 + ...)/(1 - x).
a(n) = 1 + floor(log_2(log_2(n))) for n>=2. - Kevin Ryde, Jan 18 2024

A346192 Decimal expansion of Sum_{k>=0} 1/(2^(2^k)*(2^(2^k) - 1)).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 09 2021

This constant is transcendental (Schwarz, 1967).

			0.58751531886028517686812633660668413283432738560133...

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, A087046, , A346190, A346191.

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

Equals Sum_{k>=2} 1/A087046(k).

A087089 Periods of logistic map intervals in order of size.

Original entry on oeis.org

1, 2, 4, 8, 3, 6, 16, 6, 5, 12, 10, 12, 32, 12, 10, 4, 5, 7, 24, 24, 20, 20, 8, 8, 7, 24, 10, 14, 9, 64, 24, 10, 16, 14, 20, 8, 18, 14, 9, 48, 48, 14, 20, 40, 6, 48, 16, 9, 40, 40, 7, 18, 48, 16, 20, 16, 28, 18, 9, 28

Offset: 1

Enrico T. Federighi (rico125162(AT)aol.com), Aug 11 2003

nonn

The region of stability for period 8 after the point where period 4 splits in two is from 3.5440903596 to 3.5644072661 or a width of .0203169065. The period 3 cycle starts at 3.8284271247 = 1+sqrt(8) and ends at 3.8414990075, a width of .0130718828. This is less than that of period 8 so it follows it in sequence. The logistic map is just the real part of the Mandelbrot set.

The equation f(x)=a*x(1-x), f2(x)=f(f(x)) has a period 3 oscillation whenever 3.82843

For references see A087046.

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A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

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A220294 a(n) = 1 - 2^(2^n) + 2^(2^(n+1)).

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A220161 a(n) = 1 + 2^(2^n) + 2^(2^(n+1)).

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A111403 a(n) = f(f(n+1)) - f(f(n)), where f(m) = 2^m.

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Maple

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A211667 Number of iterations sqrt(sqrt(sqrt(...(n)...))) such that the result is < 2.

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A346192 Decimal expansion of Sum_{k>=0} 1/(2^(2^k)*(2^(2^k) - 1)).

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A087089 Periods of logistic map intervals in order of size.

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