A246682 Permutation of natural numbers: a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

0, 1, 2, 4, 3, 6, 5, 9, 7, 8, 11, 12, 31, 10, 13, 16, 127, 14, 709, 15, 19, 20, 5381, 21, 17, 46, 23, 18, 52711, 22, 648391, 26, 29, 166, 41, 24, 9737333, 858, 71, 25, 174440041, 30, 3657500101, 32, 37, 6186

Offset

1,3

Comments

Note the indexing: the domain starts from 1, while the range includes also zero.

Has an infinite number of infinite cycles. See comments at A246681.

Links

Table of n, a(n) for n=1..46.

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

As a composition of related permutations:

a(n) = A246378(A243071(n)).

Other identities.

For all n >= 1 the following holds:

a(A000040(n)) = A007097(n-1). [Maps primes to the iterates of primes].

A049076(a(A000040(n))) = n. [Follows from above].

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 A246378 & A246380 have the same property].

Prog

(PARI)
default(primelimit, (2^31)+(2^30));
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)};
A246682(n) = if(n < 3, n-1, if(!(n%2), A002808(A246682(n/2)), prime(A246682(A064989(n)))));
for(n=1, 46, write("b246682.txt", n, " ", A246682(n)));
(Scheme, two variants)
(definec (A246682 n) (cond ((<= n 2) (- n 1)) ((even? n) (A002808 (A246682 (/ n 2)))) (else (A000040 (A246682 (A064989 n))))))
(define (A246682 n) (A246378 (A243071 n)))

Crossrefs

Inverse: A246681.

Similar or related permutations: A246376, A246378, A243071, A246368, A064216, A246380.

Cf. A000035, A000040, A002808, A007097, A010051, A064989, A049076, A078442.

Sequence in context: A354374 A354453 A281120 * A143691 A129767 A119618

Adjacent sequences: A246679 A246680 A246681 * A246683 A246684 A246685

Keyword

nonn

Author

Antti Karttunen, Sep 01 2014

Status

approved