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%I #75 Dec 08 2022 11:55:23
%S 1,2,6,12,30,60,180,360,210,420,1260,2520,6300,12600,37800,75600,2310,
%T 4620,13860,27720,69300,138600,415800,831600,485100,970200,2910600,
%U 5821200,14553000,29106000,87318000,174636000,30030,60060,180180,360360,900900,1801800,5405400,10810800,6306300,12612600,37837800,75675600
%N If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).
%C 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.
%C 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:
%C 1
%C |
%C ...................2....................
%C 6 12
%C 30......../ \........60 180......../ \......360
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 210 420 1260 2520 6300 12600 37800 75600
%C etc.
%H Antti Karttunen, <a href="/A283477/b283477.txt">Table of n, a(n) for n = 0..4095</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
%F Other identities. For all n >= 0 (or for n >= 1):
%F a(2n+1) = 2*a(2n).
%F a(n) = A108951(A019565(n)).
%F A097248(a(n)) = A283475(n).
%F A007814(a(n)) = A051903(a(n)) = A000120(n).
%F A001221(a(n)) = A070939(n).
%F A001222(a(n)) = A029931(n).
%F A048675(a(n)) = A005187(n).
%F A248663(a(n)) = A006068(n).
%F A090880(a(n)) = A283483(n).
%F A276075(a(n)) = A283984(n).
%F A276085(a(n)) = A283985(n).
%F A046660(a(n)) = A124757(n).
%F A056169(a(n)) = A065120(n). [seems to be]
%F A005361(a(n)) = A284001(n).
%F A072411(a(n)) = A284002(n).
%F A007913(a(n)) = A284003(n).
%F A000005(a(n)) = A284005(n).
%F A324286(a(n)) = A324287(n).
%F A276086(a(n)) = A324289(n).
%F A267263(a(n)) = A324341(n).
%F A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
%F G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - _Ilya Gutkovskiy_, Aug 19 2019
%t 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 *)
%o (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
%o (Scheme)
%o (define (A283477 n) (A108951 (A019565 n)))
%o ;; Recursive "binary tree" implementation, using memoization-macro definec:
%o (definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))
%o (Python)
%o from sympy import prime, primerange, factorint
%o from operator import mul
%o from functools import reduce
%o def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
%o def a108951(n):
%o f = factorint(n)
%o return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
%o 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_
%o def a(n): return a108951(a019565(n))
%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 22 2017
%o (Python)
%o from sympy import primorial
%o from math import prod
%o 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
%Y Cf. A129912 (same terms, but sorted into ascending order).
%Y 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.
%Y Cf. A005940, A052330, A322827, A323505 for other similar trees.
%Y Cf. also A260443.
%K nonn
%O 0,2
%A _Antti Karttunen_, Mar 16 2017
%E More formulas and the binary tree illustration added by _Antti Karttunen_, Mar 19 2017
%E Four more linking formulas added by _Antti Karttunen_, Feb 25 2019