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Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].
5

%I #14 Nov 02 2014 12:18:37

%S 1,2,4,3,7,6,11,5,12,10,17,9,23,16,19,8,29,18,35,15,28,25,41,14,31,34,

%T 30,24,51,27,59,13,44,43,47,26,67,52,58,22,77,42,83,38,49,61,89,21,70,

%U 46,73,53,99,45,69,37,88,75,111,40,119,85,72,20,94,64,127,63,103,68,137,39,143,97,79,78,106,87,151,36

%N Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].

%C Terms at a(2^n) are: 1, 2, 3, 5, 8, 13, 20, 32, 48, 71, 105, 156, 236, 354, 542, 815, 1228, ...

%C Fixed points begin as: 1, 2, 6, 10, 18, 42, 92, 26372, ...

%H Antti Karttunen, <a href="/A246165/b246165.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>

%e By definition, a(1) = 1.

%e After that, for n = 2, when its prime factorization is shifted once right, results A064989(2) = 1, so we select the 1st of still unused positive natural numbers, which is 2, thus a(2) = 2.

%e For n = 3 = p_2 (3 is the second prime), when its prime factorization is shifted once right, results A064989(3) = 2 = p_1, so we select 2nd of still unused numbers, which is 4, thus a(3) = 4.

%e For n = 4, like for all powers of two, the result of right shifting is 1, so we select the smallest still unused number, which is 3, thus a(4) = 3.

%e For n = 5 = p_3, A064989(5) = 3 = p_2, so we select the 3rd smallest still unused number from [5, 6, 7, 8, ...] which is 7, thus a(5) = 7.

%o (Scheme, with defineperm1-macro from _Antti Karttunen_'s IntSeq-library)

%o (defineperm1 (A246165 n) (if (<= n 1) n (let loop ((i 1) (the-n-th-one (- (A064989 n) 1))) (cond ((not-lte? (A246166 i) n) (if (zero? the-n-th-one) i (loop (+ i 1) (- the-n-th-one 1)))) (else (loop (+ i 1) the-n-th-one))))))

%o ;; We consider a > b (i.e. not less than or equivalent to b) also in case a is nil.

%o ;; (Which is needed when using the stateful caching system used by defineperm1-macro):

%o (define (not-lte? a b) (cond ((not (number? a)) #t) (else (> a b))))

%Y Inverse: A246166.

%Y Similar permutations: A119435, A126917.

%Y Cf. A064989.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 17 2014