

A163511


a(0)=1. a(n) = p(A000120(n)) * product{m=1 to A000120(n)} p(m)^A163510(n,m), where p(m) is the mth prime.


57



1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 25, 12, 15, 10, 7, 32, 81, 54, 125, 36, 75, 50, 49, 24, 45, 30, 35, 20, 21, 14, 11, 64, 243, 162, 625, 108, 375, 250, 343, 72, 225, 150, 245, 100, 147, 98, 121, 48, 135, 90, 175, 60, 105, 70, 77, 40, 63, 42, 55, 28, 33, 22, 13, 128
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OFFSET

0,2


COMMENTS

This is a permutation of the positive integers.
From Antti Karttunen, Jun 20 2014: (Start)
Note the indexing: the domain starts from 0, while the range excludes zero, thus this is neither a bijection on the set of nonnegative integers nor on the set of positive natural numbers, but a bijection from the former set to the latter.
Apart from that discrepancy, this could be viewed as yet another "entanglement permutation" where the two complementary pairs to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with the complementary pair even numbers (taken straight) and odd numbers in the order they appear in A003961: (A005843/A003961). See also A246375 which has almost the same recurrence.
Note how the even bisection halved gives the same sequence back. (For a(0)=1, take ceiling of 1/2).
(End)


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences that are permutations of the natural numbers


FORMULA

For n >= 1, a(2n) is even, a(2n+1) is odd. a(2^k) = 2^(k+1), for all k >= 0.
From Antti Karttunen, Jun 20 2014: (Start)
a(0) = 1, a(1) = 2, a(2n) = 2*a(n), a(2n+1) = A003961(a(n)).
As a more general observation about the parity, we have:
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [This permutation preserves the 2adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, A055396(a(n)) = A091090(n) = A007814(n+1) + 1  A036987(n).
For n >= 1, a(A000225(n)) = A000040(n).
(End)
From Antti Karttunen, Oct 11 2014: (Start)
As a composition of related permutations:
a(n) = A005940(1+A054429(n)).
a(n) = A064216(A245612(n))
a(n) = A246681(A246378(n)).
Also, for all n >= 0, it holds that:
A161511(n) = A243503(a(n)).
A243499(n) = A243504(a(n)).
(End)


EXAMPLE

For n=3, whose binary representation is "11", we have A000120(3)=2, with A163510(3,1) = A163510(3,2) = 0, thus a(3) = p(2) * p(1)^0 * p(2)^0 = 3*1*1 = 3.
For n=9, "1001" in binary, we have A000120(9)=2, with A163510(9,1) = 0 and A163510(9,2) = 2, thus a(9) = p(2) * p(1)^0 * p(2)^2 = 3*1*9 = 27.
For n=10, "1010" in binary, we have A000120(10)=2, with A163510(10,1) = 1 and A163510(10,2) = 1, thus a(10) = p(2) * p(1)^1 * p(2)^1 = 3*2*3 = 18.
For n=15, "1111" in binary, we have A000120(15)=4, with A163510(15,1) = A163510(15,2) = A163510(15,3) = A163510(15,4) = 0, thus a(15) = p(4) * p(1)^0 * p(2)^0 * p(3)^0 * p(4)^0 = 7*1*1*1*1 = 7.


MATHEMATICA

f[n_] := Reverse@ Map[Ceiling[(Length@ #  1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~
Table[Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}]][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2016 *)


PROG

(Scheme, with memoizing definecmacro from Antti Karttunen's IntSeqlibrary)
;; Version based on given recurrence:
(definec (A163511 n) (cond ((<= n 1) (+ n 1)) ((even? n) (* 2 (A163511 (/ n 2)))) (else (A003961 (A163511 (/ ( n 1) 2))))))
;; Version based on Quet's original formula:
(define (A163511 n) (if (zero? n) 1 (let ((w (A000120 n))) (let loop ((p (A000040 w)) (m w)) (cond ((zero? m) p) (else (loop (* p (expt (A000040 m) (A163510 (+ (A000788 ( n 1)) m)))) ( m 1))))))))
;; Antti Karttunen, Jun 20 2014


CROSSREFS

Inverse: A243071.
Cf. A000040, A000120, A000225, A000788, A003961, A005940, A007814, A054429, A055396, A064216, A135523, A163510, A245605, A245612, A246375, A246378, A246681, A161511, A243499, A243503, A243504.
Sequence in context: A182944 A269385 A252755 * A285322 A129593 A279355
Adjacent sequences: A163508 A163509 A163510 * A163512 A163513 A163514


KEYWORD

base,nonn,look


AUTHOR

Leroy Quet, Jul 29 2009


EXTENSIONS

More terms computed and examples added by Antti Karttunen, Jun 20 2014


STATUS

approved



