A237739 -id:A237739 - cp's OEIS frontend

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

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

A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 6, 21, 11, 13, 8, 31, 14, 15, 10, 87, 29, 37, 17, 49, 19, 25, 12, 141, 42, 51, 20, 63, 22, 33, 16, 517, 112, 133, 40, 189, 50, 69, 24, 259, 64, 75, 27, 111, 35, 43, 18, 925, 177, 211, 56, 267, 66, 79, 28, 339, 83, 93, 30, 159, 45, 67, 23, 4129, 618, 685, 143, 855, 167, 201, 54, 1275, 234, 261, 65, 391, 90, 105, 34

Offset: 1

Clark Kimberling, Dec 24 2010

A permutation of the positive integers. See the comment at A183079.

			Top 6 levels of the binary tree:
                                     1
                                     |
                  ...................2...................
                 3                                       4
       7......../ \........5                   9......../ \........6
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  21       11         13       8          31       14         15       10
87  29   37  17     49  19   25 12     141  42   51  20     63  22   33  16
...
From the level 3 to the level 4: 3 --> (7,5) and 4 --> (9,6).

Antti Karttunen, Table of n, a(n) for n = 1..512
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A257690.
Cf. A000959, A050505.
Cf. A257726 (similar permutation with a slightly different definition, resulting the first differing term at n=9, where a(9) = 21, while A257726(9) = 13), A257735 - A257738.
Cf. A183079, A237739 (other similar permutations).

Let L(n) = A000959(n), the n-th lucky number.
Let U(n) = A050505(n), the n-th unlucky numbers.
The tree-array T(n,k) is then given by rows:
T(0,0) = 1; T(1,0) = 2;
T(n,2j) = L(T(n-1),j);
T(n,2j+1) = U(T(n-1),j);
for j = 0, 1, ..., 2^(n-1) - 1, n >= 2.
a(1) = 1; a(2n) = A050505(a(n)), a(2n+1) = A000959(a(n+1)). - Antti Karttunen, May 09 2015

Added a formula to the Name field and more terms, edited Example section - Antti Karttunen, May 09 2015

A269848 a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A064989(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 7, 9, 31, 10, 127, 18, 13, 14, 709, 12, 5381, 15, 23, 45, 52711, 16, 17, 165, 19, 27, 648391, 21, 9737333, 22, 61, 856, 41, 20, 174440041, 6185, 197, 24, 3657500101, 34, 88362852307, 63, 29, 58644, 2428095424619, 25, 59, 26, 977, 212

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Term a(47) manually copied from A007097(15). Note that A000040(15) = 47.

Index entries for sequences that are permutations of the natural numbers

Inverse: A269847.
Cf. A000040, A007097, A002808, A064989, A065090.
Related or similar permutations: A237739, A243071, A246682, A269858.

allocatemem(2^30);
default(primelimit,4294965247);
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A269848 = n -> if(n<=2,n,if((n%2),prime(A269848(A064989(n))),A002808(A269848(n/2)-1)));
for(n=1, 52, t = A269848(n); print1(t,", "); write("b269848.txt", n, " ", t));

;; With memoization-macro definec.
(definec (A269848 n) (cond ((<= n 1) n) ((even? n) (A065090 (+ 1 (A269848 (/ n 2))))) (else (A000040 (A269848 (A064989 n))))))

a(1) = 1, a(2) = 2, for n > 2, if n is even, a(n) = A002808(a(n/2)-1), and for odd n, a(n) = A000040(a(A064989(n))).
As a composition of other permutations:
a(n) = A237739(A243071(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A007097(n). [Maps primes to the primeth recurrence.]

A269858 Permutation of natural numbers: a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A268674(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 7, 9, 31, 10, 127, 18, 13, 14, 709, 12, 5381, 15, 19, 45, 52711, 16, 17, 165, 23, 27, 648391, 21, 9737333, 22, 29, 856, 41, 20, 174440041, 6185, 61, 24, 3657500101, 28, 88362852307, 63, 43, 58644, 2428095424619, 25, 59, 26, 37, 212, 75063692618249, 34, 67, 39, 47

Offset: 1

Antti Karttunen, Mar 06 2016

nonn

Terms for prime positions copied from A007097.

Index entries for sequences that are permutations of the natural numbers

Inverse: A269857.
Cf. A000040, A007097, A065090, A268674.
Related or similar permutations: A237739, A252756, A269848.

a(1) = 1, a(2) = 2, for n > 2, if n is even, a(n) = A002808(a(n/2)-1), and for odd n, a(n) = A000040(a(A268674(n))).
As a composition of other permutations:
a(n) = A237739(A252756(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A007097(n). [Maps primes to the primeth recurrence.]

cp's OEIS Frontend

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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A183089 Tree generated by the lucky numbers: a(1) = 1; a(2n) = unlucky(a(n)), a(2n+1) = lucky(a(n+1)), where lucky = A000959, unlucky = A050505.

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A269848 a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A064989(2n+1))).

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A269858 Permutation of natural numbers: a(1) = 1, a(2n) = A065090(1+a(n)), a(2n+1) = A000040(a(A268674(2n+1))).

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