A246379 Permutation of natural numbers: a(1) = 1, a(p_n) = A003961(1+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.

1, 3, 9, 2, 21, 6, 5, 18, 4, 42, 39, 12, 11, 10, 36, 8, 15, 84, 23, 78, 24, 22, 7, 20, 72, 16, 30, 168, 47, 46, 189, 156, 48, 44, 14, 40, 17, 144, 32, 60, 45, 336, 13, 94, 92, 378, 41, 312, 96, 88, 28, 80, 25, 34, 288, 64, 120, 90, 81, 672, 133, 26, 188, 184, 756, 82, 135, 624, 192, 176, 83, 56, 49

Offset: 1

Author

Antti Karttunen, Aug 29 2014

Keywords

• nonn

Comments

Because 2 is the only even prime, it implies that, apart from 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 each odd composite (A071904) resides in a separate infinite cycle in this permutation, except 9, which is in a finite cycle (2 3 9 4).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A246380.

Similar or related permutations: A246375, A246377, A246363, A246364, A246365, A246367, A246681.

Cf. A000720, A003961, A010051, A065855, A071904.

Programs

PARI

default(primelimit, (2^31)+(2^30));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A246379(n) = if(1==n, 1, if(isprime(n), A003961(1+A246379(primepi(n))), 2*A246379(n-primepi(n)-1)));
for(n=1, 10000, write("b246379.txt", n, " ", A246379(n)));
(Scheme, with memoization-macro definec)
(definec (A246379 n) (cond ((< n 2) n) ((= 1 (A010051 n)) (A003961 (+ 1 (A246379 (A000720 n))))) (else (* 2 (A246379 (A065855 n))))))

Formula

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n) = A003961(1+a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).

As a composition of related permutations:

a(n) = A246375(A246377(n)).

Other identities. For all n > 1 the following holds:

A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246377 & A246681 have the same property].