A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 85, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5

Offset

1,4

Comments

Sum of powers of 4 dividing n.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=0} 4^k * x^(4^k) / (1 - x^(4^k)).

L.g.f.: -log(Product_{k>=0} (1 - x^(4^k))).

Dirichlet g.f.: zeta(s) / (1 - 4^(1 - s)).

From Amiram Eldar, Nov 27 2022: (Start)

Multiplicative with a(2^e) = (4^floor((e+2)/2)-1)/3, and a(p^e) = 1 for p != 2.

Sum_{k=1..n} a(k) ~ n*log_4(n) + (1/2 + (gamma - 1)/log(4))*n, where gamma is Euler's constant (A001620). (End)

Mathematica

Table[(4^(IntegerExponent[n, 4] + 1) - 1)/3, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[4^k x^(4^k)/(1 - x^(4^k)), {k, 0, Floor[Log[4, nmax]] + 1}], {x, 0, nmax}], x] // Rest

Prog

(Python)
def A323921(n): return ((1<<((~n&n-1).bit_length()&-2)+2)-1)//3 # Chai Wah Wu, Jul 09 2022
(PARI) a(n) = (4^(valuation(n, 4) + 1) - 1) / 3; \\ Michel Marcus, Jul 09 2022

Crossrefs

Cf. A001620, A038712, A080278, A088842, A234957, A235127, A339747, A339748.

Sequence in context: A069293 A347398 A333751 * A293235 A066803 A089608

Adjacent sequences: A323918 A323919 A323920 * A323922 A323923 A323924

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Dec 15 2020

Status

approved