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%I #34 Dec 13 2015 07:46:13
%S 1,2,3,4,5,8,7,6,9,14,15,24,13,26,11,10,17,20,27,34,29,80,47,48,25,32,
%T 51,124,21,44,19,12,33,74,39,54,53,98,67,76,57,104,159,624,93,404,95,
%U 120,49,50,63,64,101,152,247,342,41,38,87,174,37,62,23,16,65,56,147,244,77,188,107,90,105,374,195,324,133,170,151,142,113,92
%N "Caves of prime shift" permutation: a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1.
%C See the comments in A246676. This is otherwise similar permutation, except that after having reached an odd number 2m-1 when we have shifted the binary representation of n right k steps, here, in contrary to A246676, we don't shift the primes in the prime factorization of the even number 2m, but instead of an even number (2*a(m)), shifting it the same number (k) of positions towards larger primes, whose product is then decremented by one to get the final result.
%C From _Antti Karttunen_, Jan 18 2015: (Start)
%C This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253885 and odd numbers in their usual order: (A253885/A005408).
%C From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253885 to the parent:
%C 1
%C |
%C ...................2...................
%C 3 4
%C 5......../ \........8 7......../ \........6
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 9 14 15 24 13 26 11 10
%C 17 20 27 34 29 80 47 48 25 32 51 124 21 44 19 12
%C (End)
%H Antti Karttunen, <a href="/A246684/b246684.txt">Table of n, a(n) for n = 1..8192</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1. [Where the bivariate function A242378(k,n) changes each prime p(i) in the prime factorization of n to p(i+k), i.e., it's the result of A003961 iterated k times starting from n].
%F a(1) = 1, a(2n) = A253885(a(n)), a(2n+1) = (2*a(n+1))-1. - _Antti Karttunen_, Jan 18 2015
%F As a composition of other permutations:
%F a(n) = A250243(A249814(n)).
%F Other identities. For all n >= 1, the following holds:
%F a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back].
%F For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].
%e Consider n=30, "11110" in binary. It has to be shifted just one bit-position right that the result were an odd number 15, "1111" in binary. As 15 = 2*8-1, we use 2*a(8) = 2*6 = 12 = 2*2*3 = p_1 * p_1 * p_2 [where p_k denotes the k-th prime, A000040(k)], which we shift one step towards larger primes resulting p_2 * p_2 * p_3 = 3*3*5 = 45, thus a(30) = 45-1 = 44.
%o (PARI)
%o 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_
%o A246684(n) = { my(k=0); if(1==n, 1, while(!(n%2), n = n/2; k++); n = 2*A246684((n+1)/2); while(k>0, n = A003961(n); k--); n-1); };
%o for(n=1, 8192, write("b246684.txt", n, " ", A246684(n)));
%o (Scheme, with memoization-macro definec, two implementations)
%o (definec (A246684 n) (cond ((<= n 1) n) (else (+ -1 (A242378bi (A007814 n) (* 2 (A246684 (A003602 n)))))))) ;; Code for A242378bi given in A242378.
%o (definec (A246684 n) (cond ((<= n 1) n) ((even? n) (A253885 (A246684 (/ n 2)))) (else (+ -1 (* 2 (A246684 (/ (+ n 1) 2)))))))
%Y Inverse: A246683.
%Y Other versions: A246676, A246678.
%Y Similar or related permutations: A005940, A163511, A241909, A245606, A246278, A246375, A249814, A250243.
%Y Cf. A000040, A000051, A003602, A003961, A007814, A242378, A253885.
%Y Differs from A249814 for the first time at n=14, where a(14) = 26, while A249814(14) = 20.
%K nonn,tabf,look
%O 1,2
%A _Antti Karttunen_, Sep 06 2014
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