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A019565 If n = Sum 2^e_i, e_i distinct, then a(n) = Product prime_{e_i+1}. 73
1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 42, 35, 70, 105, 210, 11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, 13, 26, 39, 78, 65, 130, 195, 390, 91, 182, 273, 546, 455, 910, 1365, 2730, 143, 286, 429, 858, 715, 1430, 2145, 4290 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014

From Antti Karttunen, Apr 18 & 19 2017: (Start)

Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.

Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k = 0 .. ∞. As there is on average only 6/(pi^2) = 0.6079... chance that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or for that matter, between their prime factorizations.

See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.

If either of the conjectures is false (there are cycles), then sequence A285332 (nor its inverse A285331) certainly cannot be a permutation of natural numbers.

(End)

LINKS

R. Zumkeller, Table of n, a(n) for n = 0..8191

FORMULA

G.f.: prod(k>=0, 1 + prime(k+1)*x^2^k), where prime(k)=A000040(k). - Ralf Stephan, Jun 20 2003

a(n) = f(n, 1, 1) with f(x, y, z) = if x > 0 then f(floor(x/2), y*prime(z)^(x mod 2), z+1) else y. - Reinhard Zumkeller, Mar 13 2010

From Antti Karttunen, Jul 29 2015: (Start)

Other identities. For all n >= 0:

A048675(a(n)) = n.

A013928(a(n)) = A064273(n).

(End)

a(n) = a(2^x)*a(2^y)*a(2^z)... = prime(x+1)*prime(y+1)*prime(z+1)..., where n=2^x+2^y+2^z+... - Benedict W. J. Irwin, Jul 24 2016

From Antti Karttunen, Apr 18 2017: (Start)

a(n) = A097248(A260443(n)).

a(A005187(n)) = A283475(n)

A108951(a(n)) = A283477(n).

(End)

EXAMPLE

5 = 2^2+2^0, e_1 = 2, e_2 = 0, prime(2+1) = prime(3) = 5, prime(0+1) = prime(1) = 2, so a(5) = 5*2 = 10.

This sequence regarded as a triangle withs rows of lengths 1, 1, 2, 4, 8, 16, ...:

1

2

3, 6

5, 10, 15, 30

7, 14, 21, 42, 35, 70, 105, 210

11, 22, 33, 66, 55, 110, 165, 330, 77, 154, 231, 462, 385, 770, 1155, 2310, ...

- Philippe Deléham, Jun 03 2015

MAPLE

a:= proc(n) local i, m, r; m:=n; r:=1;

      for i while m>0 do if irem(m, 2, 'm')=1

        then r:=r*ithprime(i) fi od; r

    end:

seq(a(n), n=0..60);  # Alois P. Heinz, Sep 06 2014

MATHEMATICA

Do[m=1; o=1; k1=k; While[ k1>0, k2=Mod[k1, 2]; If[k2\[Equal]1, m=m*Prime[o]]; k1=(k1-k2)/ 2; o=o+1]; Print[m], {k, 0, 55}] (* Lei Zhou, Feb 15 2005 *)

Table[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2], {n, 0, 55}]  (* Michael De Vlieger, Aug 27 2016 *)

PROG

(PARI) a(n)=factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))  \\ M. F. Hasler, Mar 26 2011, updated Aug 22 2014

(Haskell)

a019565 n = product $ zipWith (^) a000040_list (a030308_row n)

-- Reinhard Zumkeller, Apr 27 2013

(Python)

from operator import mul

from functools import reduce

from sympy import prime

def A019565(n):

....return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1

# Chai Wah Wu, Dec 25 2014

(Scheme) (define (A019565 n) (let loop ((n n) (i 1) (p 1)) (cond ((zero? n) p) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (* p (A000040 i)))) (else (loop (/ n 2) (+ 1 i) p))))) ;; (Requires only the implementation of A000040 for prime numbers.) - Antti Karttunen, Apr 20 2017

CROSSREFS

Row 1 of A285321.

Cf. A101278, A054842, A007088, A030308, A000040, A013929, A005117, A110765, A064273, A246353, A283475, A283477, A285319, A285331, A285332.

Cf. A109162 (iterates).

Cf. also A048675 (a left inverse), A097248, A260443.

Cf. A285315 (numbers for which a(n) < n), A285316 (for which a(n) > n).

Cf. A276076, A276086 (analogous sequences for factorial and primorial bases).

Sequence in context: A073740 A239956 A077320 * A274608 A133477 A039653

Adjacent sequences:  A019562 A019563 A019564 * A019566 A019567 A019568

KEYWORD

nonn,look,tabf

AUTHOR

Marc LeBrun

EXTENSIONS

Definition corrected by Klaus-R. Löffler, Aug 20 2014

STATUS

approved

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Last modified May 29 22:45 EDT 2017. Contains 287257 sequences.