A065653 -id:A065653 - cp's OEIS frontend

A065652 a(0) = 0 and a(n+1) = if a(n) - 1 is new and > 0 then a(n) - 1 else a(n)*a(n) + 1 for n >= 0.

0, 1, 2, 5, 4, 3, 10, 9, 8, 7, 6, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 122, 121, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97

Offset: 0

Reinhard Zumkeller, Nov 10 2001

nonn

a(a(n)) = n: a self-inverse permutation of the natural numbers. a(x) = x iff x = A065654(k) for some k.

Cf. A065653, A065654.

a[n_] := a[n] = If[a[n - 1] > 1 && FreeQ[Array[a, n, 0], a[n - 1] - 1], a[n - 1] - 1, a[n - 1]^2 + 1]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Mar 19 2025 *)

a(A065653(k) + j) = A065653(k+1) - 1 - j for k >= 0 and 0 <= j < A065653(k+1) - A065653(k).

A102847 a(0)=1, a(n) = a(n-1)*a(n-1) + 2.

Original entry on oeis.org

1, 3, 11, 123, 15131, 228947163, 52416803445748571, 2747521283470239265968814548542043, 7548873203121950871924356140057489033996373873303512592376938613851

Offset: 0

Miklos Kristof, Feb 28 2005

The Mandelbrot-process is z:=z*z+c, where z and c is complex. In our case c=2 and the initial z is 1. The process is very quickly increasing.
Prime for a(1)=3, a(2)=11, a(4)=15131; semiprime for a(3) = 123 = 3 * 41, a(5) = 228947163 = 3 * 76315721. a(6), added by Jonathan Vos Post, has 4 prime factors. a(7) = 41 * 811^2 * 106693969 * 317171188688357726699 * 8272236925540996054440172449761. When is the next prime in the sequence? - Jonathan Vos Post, Feb 28 2005
Composite for a(8), a(9), ..., a(19). a(20) is roughly 2^909982 and its primality is unknown. - Russ Cox, Apr 02 2006

			a(2)=11, a(3)=11*11+2=123.

Bisection of A065653.

a[0]:=1: for n from 1 to 10 do a[n]:=a[n-1]^2+2 od: seq(a[n],n=0..9); # Emeric Deutsch

a[0] := 1; a[n_] := a[n - 1]^2 + 2; Table[a[n], {n, 0, 10}] (* Stefan Steinerberger, Apr 08 2006 *)
NestList[#^2+2&,1,10] (* Harvey P. Dale, Mar 27 2023 *)

a(n)=if(n<1, n==0, 2+a(n-1)^2) /* Michael Somos, Mar 25 2006 */

a(n) ~ c^(2^n), where c = 1.8249111600523655937123650418390169034... - Vaclav Kotesovec, Sep 20 2013

a(7) from Jonathan Vos Post, Feb 28 2005

a(8) from Emeric Deutsch, Jun 13 2005

A065654 Fixed points for A065652, a permutation of the natural numbers.

Original entry on oeis.org

0, 1, 2, 4, 8, 24, 80, 784, 8288, 1053024, 115519040, 2186083514944, 26210587691915648, 9556921325803348132669824, 1373760651292040932579353684066560, 182669489453303120238622839813317479832750842872064

Offset: 0

Reinhard Zumkeller, Nov 10 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 0..23

Cf. A065652, A065653.

f[n_] := f[n] = If[n < 2, n, f[n - 2]^2 + 2]; a[n_] := (f[n] + f[n + 1] - 1)/2; Array[a, 15, 0] (* Amiram Eldar, Mar 19 2025 *)

a(n) = (A065653(n) + A065653(n+1) - 1) / 2.

cp's OEIS Frontend

A065652 a(0) = 0 and a(n+1) = if a(n) - 1 is new and > 0 then a(n) - 1 else a(n)*a(n) + 1 for n >= 0.

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A102847 a(0)=1, a(n) = a(n-1)*a(n-1) + 2.

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A065654 Fixed points for A065652, a permutation of the natural numbers.

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