OFFSET
0,3
COMMENTS
From Antti Karttunen, Feb 10 2021: (Start)
This sequence can be represented as a binary tree. Each child to the left is obtained by multiplying its parent with (1+{binary weight of its breadth-first-wise index in the tree}), while each child to the right is just a clone of its parent:
1
|
...................1...................
2 1
4......../ \........2 3......../ \........1
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
8 4 6 2 9 3 4 1
16 8 12 4 18 6 8 2 27 9 12 3 16 4 5 1
etc.
(End)
This sequence and A243499 have the same set of values on intervals from 2^m to 2^(m+1) - 1 for m >= 0. - Mikhail Kurkov, Jun 18 2021 [verification needed]
FindStat provides a sequence of mappings between this sequence and A000110 starting from collection [Set partitions] (see Links section for illustration). - Mikhail Kurkov, May 20 2023 [verification needed]
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..16384
Michael De Vlieger, This sequence as a binary tree showing rows 0 <= r <= 5.
Michael De Vlieger, This sequence as a binary tree showing rows 0 <= r <= 8.
FindStat, Illustration of the sequence of mappings
FORMULA
a(2n+1) = a(n) for n >= 0.
a(2n) = (1 + A000120(n))*a(n) = A243499(2*A059894(n)) = a(n) + a(2n - 2^A007814(n)) for n > 0 with a(0) = 1.
From Mikhail Kurkov, Apr 24 2023: (Start)
a(2^m*(2n+1)) = Sum_{k=0..m} binomial(m, k)*a(2^k*n) for m >= 0, n >= 0 with a(0) = 1.
a(n) = a(f(n)) + Sum_{k=0..floor(log_2(n))-1} (1 - T(n, k))*a(f(n) + 2^k*(1 - T(n, k))) for n > 1 with a(0) = 1, a(1) = 1, where f(n) = A053645(n) and where T(n, k) = floor(n/2^k) mod 2. (End) [verification needed]
MAPLE
a:= proc(n) option remember; `if`(n=0, 1,
a(iquo(n, 2, 'd'))*`if`(d=1, 1, add(i, i=Bits[Split](n+1))))
end:
seq(a(n), n=0..120); # Alois P. Heinz, Jun 23 2021
MATHEMATICA
Array[DivisorSigma[0, Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#1] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]]]/#2 & @@ {Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ #, (1 + Count[#, 1])!} &@ IntegerDigits[#, 2] &, 89, 0] (* Michael De Vlieger, Feb 24 2021 *)
PROG
(PARI)
(PARI) A341392(n) = if(!n, 1, if(n%2, A341392((n-1)/2), (1+hammingweight(n))*A341392(n/2))); \\ Antti Karttunen, Feb 10 2021
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Mikhail Kurkov, Feb 10 2021 [verification needed]
STATUS
approved