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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, 51, 124, 21, 44, 19, 12, 33, 74, 39, 54, 53, 98, 67, 76, 57, 104, 159, 624, 93, 404, 95, 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
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OFFSET
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1,2
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COMMENTS
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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.
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).
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:
1
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...................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 51 124 21 44 19 12
(End)
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LINKS
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FORMULA
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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].
As a composition of other permutations:
Other identities. For all n >= 1, the following holds:
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].
For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].
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EXAMPLE
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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.
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PROG
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(PARI)
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
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); };
for(n=1, 8192, write("b246684.txt", n, " ", A246684(n)));
(Scheme, with memoization-macro definec, two implementations)
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CROSSREFS
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Differs from A249814 for the first time at n=14, where a(14) = 26, while A249814(14) = 20.
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KEYWORD
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AUTHOR
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STATUS
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approved
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