0,3

Shares with permutation A237058 the property that all odd numbers occur in positions given by ludic numbers (A003309: 1, 2, 3, 5, 7, 11, 13, 17, ...), while the even numbers > 0 occur in positions given by nonludic numbers (A192607: 4, 6, 8, 9, 10, 12, 14, 15, 16, ...). However, instead of placing terms into those positions in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A237058.

Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair ludic/nonludic numbers (A003309/A192607) is entangled with a complementary pair odd/even numbers (A005408/A005843).

Because 2 is the only even ludic number, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).

Antti Karttunen, Table of n, a(n) for n = 0..10000

a(0)=0, a(1)=1; thereafter, if A192490(n) = 1 [i.e., n is ludic], a(n) = 1+(2*a(A192512(n)-1)); otherwise a(n) = 2*a(A236863(n)) [where A192512 and A236863 give the number of ludic and nonludic numbers <= n, respectively].

For n=2, with 2 being the second ludic number (= A003309(2)), the value is computed as 1+(2*a(2-1)) = 1+2*a(1) = 1+2 = 3, thus a(2)=3.

For n=3, with 3 being the third ludic number (= A003309(3)), the value is computed as 1+(2*a(3-1)) = 1+2*a(2) = 1+2*3 = 7, thus a(3)=7.

For n=4, with 4 being the first nonludic number (= A192607(1)), the value is computed as 2*a(1) = 2 = a(4).

For n=5, with 5 being the fourth ludic number (= A003309(4)), the value is computed as 1+(2*a(4-1)) = 1+2*a(3) = 1+2*7 = 15 = a(5).

For n=9, with 9 being the fourth nonludic number (= A192607(4)), the value is computed as 2*a(4) = 2*2 = 4 = a(9).

nmax = 100;

T = Range[2, nmax+7];

L = {1};

While[Length[T] > 0, With[{k = First[T]},

AppendTo[L, k]; T = Drop[T, {1, -1, k}]]];

nonL = Complement[Range[Last[L]], L];

a[n_] := a[n] = Module[{k}, Which[

n < 2, n,

IntegerQ[k = FirstPosition[L, n][[1]]], 1 + 2 a[k-1],

IntegerQ[k = FirstPosition[nonL, n][[1]]], 2 a[k],

True , Print[" error: n = ", n]]];

Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 10 2021, after Ray Chandler in A003309 *)

(Haskell)

import Data.List (elemIndex); import Data.Maybe (fromJust)

a237427 = (+ 1) . fromJust . (`elemIndex` a237126_list)

(Scheme)

;; With Antti Karttunen's IntSeq-library for memoizing definec-macro.

(definec (A237427 n) (cond ((< n 2) n) ((= 1 (A192490 n)) (+ 1 (* 2 (A237427 (- (A192512 n) 1))))) (else (* 2 (A237427 (A236863 n))))))

;; Antti Karttunen, Feb 07 2014

Inverse permutation of A237126.

Similar permutations: A135141/A227413, A243287/A243288, A243343-A243346.

Cf. A003309, A192607, A192490, A192512, A236863.

Sequence in context: A227351 A246377 A260421 * A378995 A210203 A318467

Adjacent sequences: A237424 A237425 A237426 * A237428 A237429 A237430

nonn,look

Antti Karttunen and Reinhard Zumkeller, Feb 07 2014

approved