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

Author

Antti Karttunen, Sep 01 2014

Keywords

• nonn

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

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A246681.

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

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

Programs

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)))

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].