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A337985
a(n) is the exponent of the highest power of 2 dividing the n-th Bell number.
1
0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0
OFFSET
1,5
COMMENTS
Periodic with period length 12.
LINKS
Tewodros Amdeberhan, Valerio De Angelis and Victor H. Moll, Complementary Bell numbers: Arithmetical properties and Wilf's conjecture, in: Ilias S. Kotsireas and Eugene V. Zima (eds.), Advances in Combinatorics. Springer, Berlin, Heidelberg, 2013, pp. 23-56; alternative link.
István Mező and Jose L. Ramirez, Divisibility properties of the r-Bell numbers and polynomials, Journal of Number Theory, Vol. 177 (2017), pp. 136-152; alternative link.
FORMULA
a(n) = A007814(A000110(n)).
The sequence is completely determined modulo 12 (Amdeberhan et al., 2013):
1. a(n) = 1 if n == 2 (mod 12) or n == 11 (mod 12).
2. a(n) = 2 if n == 5 (mod 12) or n == 8 (mod 12).
3. a(n) = 0 otherwise.
From Chai Wah Wu, Jul 06 2022: (Start)
a(n) = a(n-3) - a(n-6) + a(n-9) for n > 9.
G.f.: x*(-x^7 - x^4 - x)/(x^9 - x^6 + x^3 - 1). (End)
EXAMPLE
a(1) = 0 since A000110(1) = 1 is odd.
a(2) = 1 since A000110(2) = 2.
a(5) = 3 since A000110(5) = 52 = 2^2 * 13.
MATHEMATICA
a[n_] := IntegerExponent[BellB[n], 2]; Array[a, 100]
PROG
(Python)
def A337985(n): return (n%3>1)*(1+((n+3)//6&1)) # Chai Wah Wu, Jul 06 2022
CROSSREFS
Sequence in context: A263456 A002284 A016424 * A108913 A237885 A341775
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 29 2021
STATUS
approved