A246377 -id:A246377 - cp's OEIS frontend

A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(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

Offset: 1

Katarzyna Matylla, Feb 13 2008

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).
From Robert G. Wilson v, Feb 16 2008: (Start)
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.
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, ..., . (End)

			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....

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

Cf. A000720, A007814, A010051, A026238, A078442.
Cf. A246346, A246347 (record positions and values).
Cf. A227413 (inverse).
Cf. A071574, A245701, A245702, A245703, A245704, A246377, A236854, A237427 for related and similar permutations.

import Data.List (genericIndex)
a135141 n = genericIndex a135141_list (n-1)
a135141_list = 1 : map f [2..] where
   f x | iprime == 0 = 2 * (a135141 $ a066246 x) + 1
       | otherwise   = 2 * (a135141 iprime)
       where iprime = a049084 x
-- Reinhard Zumkeller, Jan 29 2014

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 *)

/* 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]];

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

from sympy import isprime, primepi
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

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)

A246378 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 23, 14, 17, 12, 13, 8, 5, 26, 53, 35, 83, 24, 43, 27, 59, 21, 37, 22, 41, 15, 19, 10, 11, 39, 101, 75, 241, 51, 149, 114, 431, 36, 89, 62, 191, 40, 103, 82, 277, 33, 73, 54, 157, 34, 79, 58, 179, 25, 47, 30, 67, 18, 29, 20, 31, 56, 167, 134, 547, 102, 379, 304, 1523, 72, 233

Offset: 1

Antti Karttunen, Aug 27 2014

Contains an infinite number of infinite cycles. See comments at A246377.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A246377.
Similar or related permutations: A237126, A054429, A227413, A236854, A246375, A246380, A246682, A163511.
Cf. A000035, A000040, A002808, A010051, A071904.

A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A246378(n) = if(1==n, 1, if(!(n%2), A002808(A246378(n/2)), prime(A246378((n-1)/2))));
for(n=1, 4096, write("b246378.txt", n, " ", A246378(n)));
(Scheme, with memoizing definec-macro)
(definec (A246378 n) (cond ((< n 2) n) ((even? n) (A002808 (A246378 (/ n 2)))) (else (A000040 (A246378 (/ (- n 1) 2))))))

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.
As a composition of related permutations:
a(n) = A227413(A054429(n)).
a(n) = A236854(A227413(n)).
a(n) = A246380(A246375(n)).
a(n) = A246682(A163511(n)). [For n >= 1].
Other identities. For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246380 & A246682 have the same property].

A071574 If n = k-th prime, a(n) = 2a(k) + 1; if n = k-th nonprime, a(n) = 2a(k).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 4, 14, 12, 15, 10, 13, 8, 28, 24, 11, 30, 9, 20, 26, 16, 29, 56, 48, 22, 60, 18, 25, 40, 31, 52, 32, 58, 112, 96, 21, 44, 120, 36, 27, 50, 17, 80, 62, 104, 57, 64, 116, 224, 192, 42, 49, 88, 240, 72, 54, 100, 23, 34, 61, 160, 124, 208, 114, 128, 19

Offset: 1

Christopher Eltschka (celtschk(AT)web.de), May 31 2002

The recursion start is implicit in the rule, since the rule demands that a(1)=2*a(1). All other terms are defined through terms for smaller indices until a(1) is reached.
a(n) is a bijective mapping from the positive integers to the nonnegative integers. Given the value of a(n), you can get back to n using the following algorithm:
Start with an initial value of k=1 and write a(n) in binary representation. Then for each bit, starting with the most significant one, do the following: - if the bit is 1, replace k by the k-th prime - if the bit is 0, replace k by the k-th nonprime. After you processed the last (i.e. least significant) bit of a(n), you've got n=k.
Example: From a(n) = 12 = 1100_2, you get 1->2->3=>6=>10; a(10)=12. Here each "->" is a step due to binary digit 1; each "=>" is a step due to binary digit 0.
The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002. (At least with this sequence the identity a(n) = A010051(n) mod 2 is obvious, because each prime is mapped to an odd number and each composite to an even number. - Antti Karttunen, Apr 04 2015)
For n > 1: a(n) = 2 * a(if i > 0 then i else A066246(n) + 1) + A057427(i) with i = A049084(n). - Reinhard Zumkeller, Feb 12 2014
A237739(a(n)) = n; a(A237739(n)) = n. - Reinhard Zumkeller, Apr 30 2014

			1 is the 1st nonprime, so a(1) = 2*a(1), therefore a(1) = 0.
2 is the 1st prime, so a(2) = 2*a(1)+1 = 2*0+1 = 1.
4 is the 2nd nonprime, so a(4) = 2*a(2) = 2*1 = 2.

Inverse: A237739.
Cf. A000720 (pi), A049084, A065855, A066246.
Compare also to the permutation A246377.
Same parity: A010051, A061007, A035026, A069754.

a071574 1 = 0
a071574 n = 2 * a071574 (if j > 0 then j + 1 else a049084 n) + 1 - signum j
            where j = a066246 n
-- Reinhard Zumkeller, Feb 12 2014

a[1]=0; a[n_]:=If[PrimeQ[n],2*a[PrimePi[n]]+1,2*a[n-PrimePi[n]]];Table[a[n],{n,100}]

first(n) = my(res = vector(n), p); for(x=2, n, p=isprime(x); res[x]=2*res[x*!p-(-1)^p*primepi(x)]+p); res \\ Iain Fox, Oct 19 2018

;; With memoizing definec-macro.
(definec (A071574 n) (cond ((= 1 n) 0) ((= 1 (A010051 n)) (+ 1 (* 2 (A071574 (A000720 n))))) (else (* 2 (A071574 (+ 1 (A065855 n)))))))
;; Antti Karttunen, Apr 04 2015

a(1) = 0, and for n > 1, if A010051(n) = 1 [when n is a prime], a(n) = 1 + 2*a(A000720(n)), otherwise a(n) = 2*a(1 + A065855(n)). - Antti Karttunen, Apr 04 2015

Mathematica program completed by Harvey P. Dale, Nov 28 2024

A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2a(n-1), a(unlucky(n)) = 2a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 5, 12, 7, 16, 10, 24, 9, 14, 13, 32, 20, 48, 18, 28, 17, 26, 64, 40, 11, 96, 36, 56, 34, 52, 25, 128, 15, 80, 22, 192, 33, 72, 112, 68, 104, 50, 21, 256, 30, 160, 44, 384, 49, 66, 19, 144, 224, 136, 208, 100, 42, 512, 60, 320, 88, 768, 29, 98, 132, 38, 27, 288, 65, 448, 272, 416, 41, 200, 97, 84, 1024, 120, 37

Offset: 0

Antti Karttunen, May 06 2015

nonn

In other words, after a(0) = 0, if n is the k-th lucky number [i.e., n = A000959(k)], a(n) = 1 + 2*a(k-1); otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], a(n) = 2*a(k).

Because all lucky numbers are odd, it means that odd numbers occur in odd positions only (together with some even numbers, for each one of which there is a separate infinite cycle), while the even positions contain only even numbers.

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

Inverse: A257726.
Cf. A005408, A005843, A000959, A050505, A109497, A145649.
Related or similar permutations: A237427, A246377, A257732, A257734.
Cf. also A257690 (another similar permutation, but with a slightly different definition, resulting the first differing term at n=13, where a(13) = 9, while A257690(13) = 11).
Cf. also A257735 - A257738.

a(0) = 0; for n >= 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = 1+(2*a(A109497(n)-1)), otherwise a(n) = 2*a(n-A109497(n)). [Where A109497(n) gives the number of lucky numbers <= n.]
As a composition of other permutations. For all n >= 1:
a(n) = A246377(A257732(n)).
a(n) = A237427(A257734(n)).

Formula in name corrected by Antti Karttunen, Jan 10 2016

A246348 a(1)=1, a(p_n) = 1 + a(n), a(c_n) = 1 + a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also binary width of terms of A135141.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 4, 3, 5, 5, 4, 4, 4, 5, 4, 4, 6, 5, 6, 5, 5, 4, 5, 6, 5, 5, 7, 6, 6, 6, 7, 6, 6, 5, 6, 5, 7, 6, 6, 5, 8, 5, 7, 7, 7, 6, 8, 7, 7, 6, 7, 5, 6, 8, 7, 7, 6, 5, 9, 7, 6, 8, 8, 8, 7, 6, 9, 8, 8, 7, 7, 6, 8, 6, 7, 9, 8, 6, 8, 7, 6, 5, 10, 8, 7, 9, 9, 6, 9, 8, 7, 10

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

If n = 1, the result is 1, otherwise, if n is prime, compute the result for that prime's index (A000720 or A049084) and add one, and if n is composite, compute the result for that composite's index (A065855) and add one.

a(n) tells how many calls (including the toplevel call) are required to compute A135141(n) or A246377(n) with a simple (nonmemoized) recursive algorithm as employed for example by Robert G. Wilson v's Mathematica-program of Feb 16 2008 in A135141 or Antti Karttunen's Scheme-proram in A246377.

Antti Karttunen, Table of n, a(n) for n = 1..32998

Cf. A000040, A002808, A000720, A065855, A070939, A135141, A246346, A246347, A246369, A246370, A246377.

\\ Compute the b-files for both the positions of records (A246346) and their values (A246347) and also for A246348 (somewhat naively):
default(primelimit, (2^31)+(2^30));
A070939 = n->#binary(n)+!n \\ From M. F. Hasler
A135141(n) = if(1==n, 1, if(isprime(n), 2*A135141(primepi(n)), 1+(2*A135141(n-primepi(n)-1))));
A246348(n) = A070939(A135141(n));
prevmax = -1; i = 0; for(n=1, 123456, if((k=A135141(n)) > prevmax, prevmax = k; i++; write("b246346.txt", i, " ", n); write("b246347.txt", i, " ", k)); write("b246348.txt", n, " ", A246348(n)));
(Scheme, two versions, second being a direct recurrence employing memoizing definec-macro from Antti Karttunen's IntSeq-library)
(define (A246348 n) (A070939 (A135141 n)))
(definec (A246348 n) (cond ((= 1 n) 1) ((= 1 (A010051 n)) (+ 1 (A246348 (A000720 n)))) (else (+ 1 (A246348 (A065855 n))))))

a(1) = 1, and for n >= 1, if A010051(n)=1 [that is, when n is prime], a(n) = 1 + a(A000720(n)), otherwise a(n) = 1 + a(A065855(n)). [A000720(n) and A065855(n) tell the number of primes, and respectively, composites <= n].
a(n) = A246369(n) + A246370(n).
a(n) = A070939(A135141(n)) = 1 + floor(log_2(A135141(n))). [Sequence gives also the binary width of terms of A135141].
a(n) = A070939(A246377(n)). [Also for 0/1-swapped version of that sequence].

A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 13, 12, 21, 18, 11, 16, 25, 14, 33, 20, 15, 26, 29, 24, 17, 42, 19, 36, 53, 22, 73, 32, 43, 50, 37, 28, 45, 66, 31, 40, 57, 30, 81, 52, 27, 58, 61, 48, 49, 34, 35, 84, 117, 38, 41, 72, 87, 106, 169, 44, 213, 146, 67, 64, 65, 86, 89, 100, 91, 74, 173, 56, 149, 90, 51, 132, 101, 62, 113, 80, 23

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A246375.
Similar or related permutations: A005940, A005941, A064216, A243071, A245605, A246377, A246380.
Cf. A000035, A064989.

a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).
As a composition of related permutations:
a(n) = A246377(A246380(n)).
Other identities. For all n >= 1 the following holds:
A000035(a(n)) = A000035(n). [Like A005940 & A005941, this also preserves the parity].

A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 23, 19, 20, 21, 25, 22, 24, 26, 27, 28, 29, 34, 37, 30, 31, 32, 36, 33, 41, 35, 38, 39, 43, 40, 47, 42, 49, 52, 53, 44, 45, 46, 51, 48, 61, 57, 50, 54, 55, 59, 67, 56, 71, 64, 58, 66, 70, 72, 97, 60, 62, 63, 77, 69, 83, 65, 81

Offset: 1

Antti Karttunen, Feb 23 2015

nonn

This can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case a complementary pair ludic/nonludic numbers (A003309/A192607) is intertwined with a complementary pair prime/composite numbers (A000040/A002808).

			When n = 19 = A000040(8) [the eighth prime], we look for the value of a(8), which is 8 [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 eighth ludic number larger than 1, which is A003309(1+8) = 23, thus a(19) = 23.
When n = 20 = A002808(11) [the eleventh composite], we look for the value of a(11), which is 11 [all terms less than 19 are fixed, see above], and then take the eleventh nonludic number, which is A192607(11) = 19, thus a(20) = 19.
When n = 30 = A002808(19) [the 19th composite], we look for the value of a(19), which is 23 [see above], and then take the 23rd nonludic number, which is A192607(23) = 34, thus a(30) = 34.

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

Inverse: A255422.
Cf. A000040, A000720, A002808, A003309, A007097, A008578, A065855, A010051, A192607, A255420, A255324.
Related or similar permutations: A237126, A246377, A245703, A245704, A255407, A255408.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003309(1+a(A000720(n))), otherwise a(n) = A192607(a(A065855(n))).
As a composition of other permutations:
a(n) = A237126(A246377(n)).
Other identities.
a(A007097(n)) = A255420(n). [Maps iterates of primes to the iterates of Ludic numbers.]

A246681 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(p_n) = A003961(a(n)), a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n), and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 9, 10, 8, 14, 11, 12, 15, 18, 20, 16, 25, 28, 21, 22, 24, 30, 27, 36, 40, 32, 50, 56, 33, 42, 13, 44, 48, 60, 54, 72, 45, 80, 64, 100, 35, 112, 75, 66, 84, 26, 63, 88, 96, 120, 108, 144, 81, 90, 160, 128, 200, 70, 49, 224, 99, 150, 132, 168, 52, 126, 55, 176, 192, 240, 39

Offset: 0

Antti Karttunen, Sep 01 2014

Note the indexing: the domain starts from 0, while the range excludes zero.
Iterating a(n) from n=0 gives the sequence: 1, 2, 3, 5, 7, 9, 8, 10, 14, 18, 28, 56, 128, 156, 1344, 16524, 2706412500, ..., which is the only one-way cycle of this permutation.
Because 2 is the only even prime, it implies that, apart from a(0)=1 and a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions). This in turn implies that there exists an infinite number of infinite cycles like (... 648391 31 13 15 20 22 30 42 112 196 1350 ...) which contain just one odd composite (A071904). Apart from 9 which is in that one-way cycle, each odd composite occurs in a separate infinite two-way cycle, like 15 in the example above.

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

Inverse: A246682.
Similar or related permutations: A163511, A246377, A246379, A246367, A245821.
Cf. A000040, A000720, A003961, A007097, A010051, A065855, A071904, A078442, A049076, A055396.

a(0) = 1, a(1) = 2, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003961(a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
Other identities.
For all n >= 0, the following holds:
a(A007097(n)) = A000040(n+1). [Maps the iterates of primes to primes].
A078442(a(n)) > 0 if and only if n is in A007097. [Follows from above].
For all n >= 1, the following holds:
a(n) = A163511(A246377(n)).
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246377 & A246379 have the same property].
A055396(a(n)) = A049076(n). [An "order of primeness" is mapped to the index of the smallest prime dividing n].

A246370 a(1)=0, a(p_n) = 1 + a(n), a(c_n) = a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also number of nonleading 0-bits in the binary representation of A135141(n).

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 1, 2, 0, 3, 4, 1, 2, 1, 2, 0, 2, 3, 3, 4, 1, 2, 1, 1, 2, 0, 2, 3, 4, 3, 5, 4, 1, 2, 1, 1, 2, 2, 0, 2, 3, 3, 2, 4, 3, 5, 3, 4, 1, 2, 1, 1, 1, 2, 2, 0, 2, 3, 3, 3, 4, 2, 4, 3, 5, 3, 4, 4, 1, 2, 5, 1, 2, 1, 1, 2, 2, 0, 3, 2, 3, 3, 2, 3, 4, 2, 4, 3, 2, 5, 3, 4, 4, 1, 2, 5, 3, 1, 2, 1, 1, 1, 3, 2, 2, 0, 4, 3, 5, 2, 3, 3, 4, 2, 3, 4, 2, 4, 3, 2

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

			Consider n=30. It is the 19th composite number in A002808: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, ...
Thus we consider next n=19, which is the 8th prime in A000040: 2, 3, 5, 7, 11, 13, 17, 19, ...
So we proceed with n=8, which is the 3rd composite number, and then with n=3, which is the 2nd prime, and then with n=2 which is the 1st prime, and we have finished.
All in all, it took us 5 steps (A246348(30) = 6 = 5+1) to reach 1, and on the journey, we encountered three primes, 19, 3 and 2, thus a(30) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..32998

Cf. A000040, A002808, A000720, A065855, A080791, A135141, A246348, A246369, A246377.

a(1) = 1, and for n >= 1, if A010051(n) = 1 [i.e. when n is prime], a(n) = 1 + a(A000720(n)), otherwise a(n) = a(A065855(n)). [A000720(n) and A065855(n) tell the number of primes, and respectively, composites <= n].
a(n) = A080791(A135141(n)). [a(n) tells also the number of nonleading zeros in binary representation of A135141(n)].
a(n) = A000120(A246377(n))-1. [Respectively, one less than the number of 1-bits in 0/1-swapped version of that sequence].
a(n) = A246348(n) - A246369(n) - 1.

A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 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, 63, 203, 12, 22, 13, 17, 28, 10, 35, 78, 235, 83, 1093, 251, 18, 30, 19, 24, 31, 39, 16, 47, 67, 101, 43, 290, 107, 1283, 87, 309, 26, 41, 27, 34, 21, 42, 53, 23, 61, 88, 115, 128, 321, 57, 354, 137, 1499, 112, 349, 376, 36, 55, 1401, 38, 49, 46, 29, 56, 70, 32, 99, 81

Offset: 1

Antti Karttunen, May 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6185
Index entries for sequences that are permutations of the natural numbers

Inverse: A257732.
Cf. A000040, A000720, A000959, A002808, A010051, A050505, A065855.
Related or similar permutations: A246377, A255421, A257726, A257733.
Cf. also A032600, A255553, A255554.
Differs from A257733 for the first time at n=19, where a(19) = 63, while A257733(19) = 203.

a(1) = 1; for n > 1: if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A000959(1+a(A000720(n))), otherwise a(n) = A050505(a(A065855(n))).
As a composition of other permutations:
a(n) = A257726(A246377(n)).
a(n) = A257733(A255421(n)).

cp's OEIS Frontend

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.

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A246378 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

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A071574 If n = k-th prime, a(n) = 2*a(k) + 1; if n = k-th nonprime, a(n) = 2*a(k).

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A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2*a(n-1), a(unlucky(n)) = 2*a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.

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A246348 a(1)=1, a(p_n) = 1 + a(n), a(c_n) = 1 + a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also binary width of terms of A135141.

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A246376 Permutation of natural numbers: a(1) = 1, a(2n) = 2 * a(n), a(2n-1) = 1 + (2 * a(A064989(2n-1)-1)).

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A255421 Permutation of natural numbers: a(1) = 1, a(p_n) = ludic(1+a(n)), a(c_n) = nonludic(a(n)), where p_n = n-th prime, c_n = n-th composite number and ludic = A003309, nonludic = A192607.

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A246681 Permutation of natural numbers: a(0) = 1, a(1) = 2, a(p_n) = A003961(a(n)), a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n), and A003961(n) shifts the prime factorization of n one step towards larger primes.

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A135141 a(1)=1, a(p_n)=2a(n), a(c_n)=2a(n)+1, where p_n = n-th prime, c_n = n-th composite number.

A071574 If n = k-th prime, a(n) = 2a(k) + 1; if n = k-th nonprime, a(n) = 2a(k).

A257725 Permutation of natural numbers: a(0) = 0, a(lucky(n)) = 1 + 2a(n-1), a(unlucky(n)) = 2a(n), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505.