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A283477
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If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).
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41
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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
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OFFSET
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0,2
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COMMENTS
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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:
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...................2....................
6 12
30......../ \........60 180......../ \......360
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
210 420 1260 2520 6300 12600 37800 75600
etc.
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LINKS
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FORMULA
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Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
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MATHEMATICA
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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 *)
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PROG
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(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
(Scheme)
;; Recursive "binary tree" implementation, using memoization-macro definec:
(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))
(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
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017
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STATUS
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approved
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