A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

1, 4, 2, 9, 7, 6, 23, 16, 3, 14, 13, 12, 43, 35, 17, 26, 37, 8, 101, 24, 5, 22, 19, 21, 53, 62, 83, 51, 79, 27, 233, 39, 191, 54, 149, 15, 103, 134, 11, 36, 47, 10, 151, 34, 41, 30, 29, 33, 73, 75, 241, 86, 113, 114, 89, 72, 1153, 108, 443, 40, 593, 296, 547, 56, 167, 245, 173, 76, 563, 194, 1553, 25

Offset: 1

Author

Antti Karttunen, Aug 29 2014

Keywords

• nonn

Comments

Has an infinite number of infinite cycles. See comments in A246379.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..3098
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A246379.

Similar or related permutations: A246376, A246378, A246363, A246364, A246366, A246368, A064216, A246682.

Cf. A000035, A000040, A002808, A010051, A064989.

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)};
A246380(n) = if(1==n, 1, if(!(n%2), A002808(A246380(n/2)), prime(A246380(A064989(n)-1))));
for(n=1, 3098, write("b246380.txt", n, " ", A246380(n)));
(Scheme, with memoization-macro definec)
(definec (A246380 n) (cond ((< n 2) n) ((even? n) (A002808 (A246380 (/ n 2)))) (else (A000040 (A246380 (- (A064989 n) 1))))))

Formula

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-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(A246376(n)).

Other identities. 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 & A246682 have the same property].