A236863 -id:A236863 - cp's OEIS frontend

A237427 a(0)=0, a(1)=1; thereafter, if n is k-th ludic number [i.e., n = A003309(k)], a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = 2a(k).

0, 1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 62, 24, 26, 20, 29, 56, 9, 16, 22, 120, 61, 124, 48, 52, 40, 58, 112, 18, 63, 32, 44, 240, 25, 122, 27, 248, 96, 104, 21, 80, 116, 224, 36, 126, 57, 64, 88, 480, 50, 244, 54, 496, 17, 192, 208, 42

Offset: 0

Antti Karttunen and Reinhard Zumkeller, Feb 07 2014

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).

			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).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation of A237126.
Similar permutations: A135141/A227413, A243287/A243288, A243343-A243346.
Cf. A003309, A192607, A192490, A192512, A236863.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a237427 = (+ 1) . fromJust . (`elemIndex` a237126_list)

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 *)

;; 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

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].

A235491 Self-inverse permutation of natural numbers: complementary pair ludic/nonludic numbers (A003309/A192607) entangled with the same pair in the opposite order, nonludic/ludic. See Formula.

Original entry on oeis.org

0, 1, 4, 9, 2, 16, 7, 6, 25, 3, 61, 26, 17, 14, 13, 115, 5, 12, 359, 119, 67, 47, 43, 36, 791, 8, 11, 41, 3017, 81, 811, 407, 247, 227, 179, 7525, 23, 38, 37, 221, 34015, 27, 503, 22, 7765, 3509, 1943, 21, 1777, 1333, 93625, 97, 193, 146, 181, 1717, 486721, 121, 4493, 91, 96839, 10, 40217, 20813, 89

Offset: 0

Antti Karttunen, Feb 07 2014

nonn

The permutation is self-inverse (an involution), meaning that a(a(n)) = n for all n.

			For n=2, with 2 being the second ludic number (= A003309(4)), the value is computed as nonludic(a(2-1)) = nonludic(a(1)) = 4, the first nonludic number, thus a(2) = 4.
For n=5, with 5 being the fourth ludic number (= A003309(4)), the value is computed as nonludic(a(4-1)) = nonludic(a(3)) = nonludic(9) = 16, thus a(5) = 16.
For n=6, with 6 being the second nonludic number (= A192607(2)), the value is computed as ludic(a(2)+1) = ludic(4+1) = ludic(5) = 7, thus a(6) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..74 (The terms after a(32) were computed with the help of 100000 term b-file uploaded by Donovan Johnson for A003309.)
Index entries for sequences that are permutations of the natural numbers

Cf. A236854 (a similar permutation constructed from prime and composite numbers).
Cf. A237126/A237427 (entanglement permutations between ludic/nonludic <-> odd/even numbers).
Cf. also A003309, A192607, A192490, A192512, A236863.

a(0)=0, a(1)=1, and for n > 1, if n is k-th ludic number (i.e., n = A003309(k)), then a(n) = nonludic(a(k-1)); otherwise, when n is k-th nonludic number (i.e., n = A192607(k)), then a(n) = ludic(a(k)+1), where ludic numbers are given by A003309, and nonludic numbers by A192607.

a(0)=0, a(1)=1, and for n > 1, if A192490(n)=1 (n is ludic) a(n) = A192607(a(A192512(n)-1)); otherwise (n is nonludic), a(n) = A003309(1+(a(A236863(n)))).

A255422 Permutation of natural numbers: a(1) = 1 and for n > 1, if n is k-th ludic number larger than 1 [i.e., n = A003309(k+1)], a(n) = nthprime(a(k)), otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = nthcomposite(a(k)), where nthcomposite = A002808, nthprime = A000040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 19, 25, 23, 26, 27, 28, 29, 32, 33, 34, 36, 30, 38, 35, 31, 39, 40, 42, 37, 44, 41, 48, 49, 50, 43, 52, 45, 55, 51, 46, 47, 56, 57, 60, 54, 63, 58, 68, 53, 69, 70, 62, 74, 64, 59, 77, 72, 65, 61, 66, 78, 80, 84, 76, 71, 87, 81

Offset: 1

Antti Karttunen, Feb 23 2015

The graph has a comet appearance. - Daniel Forgues, Dec 15 2015

			When n = 19 = A192607(11) [the eleventh nonludic number], we look for the value of a(11), which is 11 [all terms less than 19 are fixed because the beginnings of A003309 and A008578 coincide up to A003309(8) = A008578(8) = 17], and then take the eleventh composite number, which is A002808(11) = 20, thus a(19) = 20.
When n = 25 = A003309(10) = A003309(1+9) [the tenth ludic number, and ninth after one], we look for the value of a(9), which is 9 [all terms less than 19 are fixed, see above], and then take the ninth prime number, which is A000040(9) = 23, thus a(25) = 23.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences that are permutations of the natural numbers

Inverse: A255421.
Cf. A000040, A002808, A003309, A192490, A192512, A192607, A236863.
Related or similar permutations: A237427, A246378, A245703, A245704 (compare the scatterplots), A255407, A255408.

a(1)=1; and for n > 1, if A192490(n) = 1 [i.e., n is ludic], a(n) = A000040(a(A192512(n)-1)), otherwise a(n) = A002808(a(A236863(n))) [where A192512 and A236863 give the number of ludic and nonludic numbers <= n, respectively].

As a composition of other permutations: a(n) = A246378(A237427(n)).

A266638 a(1) = 1, a(ludic(n)) = (ludic(3+a(n-1))-1)/2, a(nonludic(n)) = A266410(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and A266410 = numbers n such that 2n+1 is nonludic.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 9, 10, 13, 8, 16, 12, 15, 19, 22, 11, 27, 17, 31, 25, 29, 18, 36, 20, 40, 24, 49, 26, 32, 54, 46, 51, 34, 62, 37, 14, 68, 43, 81, 35, 47, 23, 55, 88, 76, 33, 83, 58, 99, 64, 28, 44, 107, 72, 127, 61, 77, 42, 91, 53, 136, 121, 56, 130, 94, 21, 151, 101, 50, 65, 73, 161, 114, 189, 98, 38

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A266637.
Cf. A003309, A192490, A192512, A236863, A266409, A266410.
Related or similar permutations: A237427, A266418.

a(1) = 1; for n > 1, if A192490(n) = 1 [when n is one of Ludic numbers, A003309] a(n) = A266409(1+a(A192512(n)-1)), otherwise a(n) = A266410(a(A236863(n))).
As a composition of related permutations:
a(n) = A266418(A237427(n)).

A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 203, 12, 22, 17, 63, 28, 13, 10, 35, 78, 235, 251, 18, 30, 24, 83, 39, 19, 1093, 16, 47, 101, 31, 290, 67, 309, 26, 41, 43, 34, 107, 53, 27, 1283, 87, 23, 61, 128, 42, 354, 88, 376, 21, 36, 55, 57, 46, 137, 115, 70, 38, 1499, 321, 112, 32, 81, 161, 56, 1401, 430, 113, 454, 29, 48, 49

Offset: 1

Antti Karttunen, May 06 2015

Antti Karttunen, Table of n, a(n) for n = 1..8587
Index entries for sequences that are permutations of the natural numbers

Inverse: A257734.
Cf. A000959, A050505, A003309, A192607, A192490, A192512, A236863.
Related or similar permutations: A237427, A255422, A257726, A257731.
Cf. also A256486, A256487.
Differs from A257731 for the first time at n=19, where a(19) = 203, while A257731(19) = 63.

a(1) = 1; for n > 1: if A192490(n) = 1 [i.e., if n is ludic], then a(n) = A000959(1+a(A192512(n)-1)), otherwise a(n) = A050505(a(A236863(n))).
As a composition of other permutations:
a(n) = A257731(A255422(n)).
a(n) = A257726(A237427(n)).

cp's OEIS Frontend

A237427 a(0)=0, a(1)=1; thereafter, if n is k-th ludic number [i.e., n = A003309(k)], a(n) = 1 + (2*a(k-1)); otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = 2*a(k).

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A235491 Self-inverse permutation of natural numbers: complementary pair ludic/nonludic numbers (A003309/A192607) entangled with the same pair in the opposite order, nonludic/ludic. See Formula.

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A255422 Permutation of natural numbers: a(1) = 1 and for n > 1, if n is k-th ludic number larger than 1 [i.e., n = A003309(k+1)], a(n) = nthprime(a(k)), otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = nthcomposite(a(k)), where nthcomposite = A002808, nthprime = A000040.

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A266638 a(1) = 1, a(ludic(n)) = (ludic(3+a(n-1))-1)/2, a(nonludic(n)) = A266410(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and A266410 = numbers n such that 2n+1 is nonludic.

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A257733 Permutation of natural numbers: a(1) = 1, a(ludic(n)) = lucky(1+a(n-1)), a(nonludic(n)) = unlucky(a(n)), where ludic(n) = n-th ludic number A003309, nonludic(n) = n-th nonludic number A192607 and lucky = A000959, unlucky = A050505.

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A237427 a(0)=0, a(1)=1; thereafter, if n is k-th ludic number [i.e., n = A003309(k)], a(n) = 1 + (2a(k-1)); otherwise, when n is k-th nonludic number [i.e., n = A192607(k)], a(n) = 2a(k).