A135141 - OEIS

A135141

a(1)=1, a(p_n)=2*a(n), a(c_n)=2*a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

1, 2, 4, 3, 8, 5, 6, 9, 7, 17, 16, 11, 10, 13, 19, 15, 12, 35, 18, 33, 23, 21, 14, 27, 39, 31, 25, 71, 34, 37, 32, 67, 47, 43, 29, 55, 22, 79, 63, 51, 20, 143, 26, 69, 75, 65, 38, 135, 95, 87, 59, 111, 30, 45, 159, 127, 103, 41, 24, 287, 70, 53, 139, 151, 131, 77, 36, 271, 191

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A permutation of the positive integers, related to A078442.

a(p) is even when p is prime and is divisible by 2^(prime order of p).

What is the length of the cycle containing 10? Is it infinite? The cycle begins 10, 17, 12, 11, 16, 15, 19, 18, 35, 29, 34, 43, 26, 31, 32, 67, 36, 55, 159, 1055, 441, 563, 100, 447, 7935, 274726911, 1013992070762272391167, ... Implementation in Mmca: NestList[a(AT)# &, 10, 26] Furthermore, it appears that any non-single-digit number has an infinite cycle. (* Robert G. Wilson v, Feb 16 2008 *)

Records: 1, 2, 4, 8, 9, 17, 19, 35, 39, 71, 79, 143, 159, 287, 319, 575, 639, 1151, 1279, 2303, 2559, 4607, 5119, 9215, 10239, 18431, 20479, 36863, 40959, 73727, 81919, 147455, 163839, 294911, 327679, 589823, 655359, ..., . (* Robert G. Wilson v, Feb 16 2008 *)

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Antti Karttunen, Entanglement Permutations, 2016-2017

Index entries for sequences computed from indices in prime factorization

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = 2*A135141((A049084(n))*chip + A066246(n)*(1-chip)) + 1 - chip, where chip = A010051(n). - Reinhard Zumkeller, Jan 29 2014

From Antti Karttunen, Dec 09 2019: (Start)

A007814(a(n)) = A078442(n).

A070939(a(n)) = A246348(n).

A080791(a(n)) = A246370(n).

A054429(a(n)) = A246377(n).

A245702(a(n)) = A245703(n).

a(A245704(n)) = A245701(n).

(End)

EXAMPLE

a(20) = 33 = 2*16 + 1 because 20 is 11th composite and a(11)=16. Or, a(20)=33=100001(bin). In other words it is a composite number, its index is a prime number, whose index is a prime....

MATHEMATICA

a[1] = 1; a[n_] := If[PrimeQ@n, 2*a[PrimePi[n]], 2*a[n - 1 - PrimePi@n] + 1]; Array[a, 69] (* Robert G. Wilson v, Feb 16 2008 *)

PROG

(Maxima) Let pc = prime count (which prime it is), cc = composite count:

pc[1]:0;

cc[1]:0;

pc[2]:1;

cc[4]:1;

pc[n]:=if primep(n) then 1+pc[prev_prime(n)] else 0;

cc[n]:=if primep(n) then 0 else if primep(n-1) then 1+cc[n-2] else 1+cc[n-1];

a[1]:1;

a[n]:=if primep(n) then 2*a[pc[n]] else 1+2*a[cc[n]];