A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

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

Offset: 0

Author

Antti Karttunen, Mar 16 2017

Keywords

• nonn

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
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to primorial numbers

Crossrefs

Cf. A129912 (same terms, but 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, A324287, A324289, A324341, A324342, A324343.

Cf. A005940, A052330, A322827, A323505 for other similar trees.

Cf. also A260443.

Programs

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 *)

PARI

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

Python

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
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 range(101)]) # Indranil Ghosh, Jun 22 2017

Python

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

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)))))))

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).

A324286(a(n)) = A324287(n).

A276086(a(n)) = A324289(n).

A267263(a(n)) = A324341(n).

A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]

G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

Extensions

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

Four more linking formulas added by Antti Karttunen, Feb 25 2019