A135141 - OEIS

%I #48 Dec 09 2019 07:27:47

%S 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,

%T 25,71,34,37,32,67,47,43,29,55,22,79,63,51,20,143,26,69,75,65,38,135,

%U 95,87,59,111,30,45,159,127,103,41,24,287,70,53,139,151,131,77,36,271,191

%N 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.

%C A permutation of the positive integers, related to A078442.

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

%C 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 *)

%C 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 *)

%H Robert G. Wilson v, <a href="/A135141/b135141.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A135141/a135141.pdf">Entanglement Permutations</a>, 2016-2017

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = 2*A135141((A049084(n))*chip + A066246(n)*(1-chip)) + 1 - chip, where chip = A010051(n). - _Reinhard Zumkeller_, Jan 29 2014

%F From _Antti Karttunen_, Dec 09 2019: (Start)

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

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

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

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

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

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

%F (End)

%e 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....

%t 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 *)

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

%o pc[1]:0;

%o cc[1]:0;

%o pc[2]:1;

%o cc[4]:1;

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

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

%o a[1]:1;

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

%o (Haskell)

%o import Data.List (genericIndex)

%o a135141 n = genericIndex a135141_list (n-1)

%o a135141_list = 1 : map f [2..] where

%o    f x | iprime == 0 = 2 * (a135141 $ a066246 x) + 1

%o        | otherwise   = 2 * (a135141 iprime)

%o        where iprime = a049084 x

%o -- _Reinhard Zumkeller_, Jan 29 2014

%o (Python)

%o from sympy import isprime, primepi

%o def a(n): return 1 if n==1 else 2*a(primepi(n)) if isprime(n) else 2*a(n - 1 - primepi(n)) + 1 # _Indranil Ghosh_, Jun 11 2017, after Mathematica code

%o (PARI) A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1)))); \\ _Antti Karttunen_, Dec 09 2019

%Y Cf. A000720, A007814, A010051, A026238, A078442.

%Y Cf. A246346, A246347 (record positions and values).

%Y Cf. A227413 (inverse).

%Y Cf. A071574, A245701, A245702, A245703, A245704, A246377, A236854, A237427 for related and similar permutations.

%K nonn,look

%O 1,2

%A _Katarzyna Matylla_, Feb 13 2008