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A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek). 18
1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.

This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:

                                      1

                                      |

                   ...................2....................

                  6                                       12

       30......../ \........60                 180......../ \......360

       / \                 / \                 / \                 / \

      /   \               /   \               /   \               /   \

     /     \             /     \             /     \             /     \

  210       420      1260       2520     6300       12600   37800       75600

etc.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..4095

FORMULA

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).

Other identities. For all n >= 0 (or for n >= 1):

a(2n+1) = 2*a(2n).

a(n) = A108951(A019565(n)).

A097248(a(n)) = A283475(n).

A007814(a(n)) = A051903(a(n)) = A000120(n).

A001221(a(n)) = A070939(n).

A001222(a(n)) = A029931(n).

A048675(a(n)) = A005187(n).

A248663(a(n)) = A006068(n).

A090880(a(n)) = A283483(n).

A276075(a(n)) = A283984(n).

A276085(a(n)) = A283985(n).

A046660(a(n)) = A124757(n).

A056169(a(n)) = A065120(n). [seems to be]

A005361(a(n)) = A284001(n).

A072411(a(n)) = A284002(n).

A007913(a(n)) = A284003(n).

A000005(a(n)) = A284005(n).

MATHEMATICA

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

PROG

(PARI)

A034386(n) = prod(i=1, primepi(n), prime(i));

A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From Charles R Greathouse IV, Jun 28 2015

A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler

A283477(n) = A108951(A019565(n));

(PARI)

\\ A simpler implementation.

A002110(n) = prod(i=1, n, prime(i));

A030308(n, k) = bittest(n, k);

A283477(n) = prod(i=0, #binary(n), if(0==A030308(n, i), 1, A030308(n, i)*A002110(1+i)));

(Scheme)

(define (A283477 n) (A108951 (A019565 n)))

;; Recursive "binary tree" implementation, using memoization-macro definec:

(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

(Python)

from sympy import prime, primerange, factorint

from operator import mul

def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])

def a108951(n):

    f = factorint(n)

    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])

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 # after Chai Wah Wu

def a(n): return a108951(a019565(n))

print [a(n) for n in xrange(101)] # Indranil Ghosh, Jun 22 2017

CROSSREFS

Cf. A129912 (sorted into ascending order).

Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005.

Cf. also A005940, A260443.

Sequence in context: A162214 A100071 A129912 * A182863 A161507 A032177

Adjacent sequences:  A283474 A283475 A283476 * A283478 A283479 A283480

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2017

EXTENSIONS

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

STATUS

approved

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Last modified December 16 01:36 EST 2017. Contains 296063 sequences.