A325803 Nonzero terms of Product_{k=0..floor(log_2(n))} (1 + A004718(floor(n/(2^k)))).

1, 2, 6, -6, 24, -18, -48, 120, 18, -72, -192, 48, -360, 720, 54, 144, -360, 384, -960, 144, -1800, 720, -2880, 5040, -54, 216, 576, -144, 1080, -2160, 1536, -384, 2880, -5760, -144, 576, 5400, -10800, 2880, -720, -17280, 8640, -25200, 40320, -162, -432, 1080

Offset: 1

Author

Mikhail Kurkov, May 22 2019

Keywords

• sign
• look

Comments

See A329893.

Links

• Mikhail Kurkov, Table of n, a(n) for n = 1..13495

Crossrefs

Cf. A000120, A001405, A004718, A284005, A325804, A329893.

Programs

Mathematica

a[n_?EvenQ] := a[n] = -a[n/2]; a[0] = 0; a[n_] := a[n] = a[(n - 1)/2] + 1; DeleteCases[Table[Product[ 1 + a[Floor[n/(2^k)]], {k, 0, Floor[Log2[n]]}], {n, 0, 200}], 0] (* Michael De Vlieger, Apr 22 2024, after Jean-François Alcover at A004718 *)

PARI

b(n) = if(n==0, 0, (-1)^(n+1)*b(n\2) + n%2); \\ A004718
f(n) = if(n==0, 1, prod(k=0, logint(n,2), 1+b(n\2^k)));
lista(nn) = for (n=0, nn, if (f(n), print1(f(n), ", "))); \\ Michel Marcus, May 26 2019

Python

from itertools import count, islice
from math import prod
def A325803_gen(): # generator of terms
    for n in count(0):
        c, s = [0]*(m:=n.bit_length()), bin(n)[2:]
        for i in range(m):
            if s[i]=='1':
                for j in range(m-i):
                    c[j] = c[j]+1
            else:
                for j in range(m-i):
                    c[j] = -c[j]
        if (k:=prod(1+d for d in c)): yield k
A325803_list = list(islice(A325803_gen(),20)) # Chai Wah Wu, Mar 03 2023

Formula

a(n) = A329893(A325804(n)). - Antti Karttunen, Dec 10 2019

Extensions

Comments and two formulas moved to A329893, which is an "uncompressed" version of this sequence. - Antti Karttunen, Dec 11 2019