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A019565 If n = Sum 2^e_i, e_i distinct, then a(n) = Product prime_{e_i+1}. 48
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

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

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 *)

Map[Times @@ Prime@ Flatten@ Position[#, 1] &, Map[Reverse, IntegerDigits[Range[0, 55], 2]]] (* 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

CROSSREFS

Cf. A101278, A054842, A007088, A030308, A000040, A013929, A005117, A110765, A064273, A246353.

Cf. also A048675 (a left inverse).

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 December 4 21:33 EST 2016. Contains 278755 sequences.