A071574 - OEIS

A071574

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

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OFFSET

1,3

COMMENTS

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

FORMULA

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

EXAMPLE

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.

MATHEMATICA

a[1] = 0 a[n_] := If[PrimeQ[n], 2*a[PrimePi[n]] + 1, 2*a[n - PrimePi[n]]]

PROG

(Haskell)

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