cp's OEIS Frontend

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

%I A323921 #27 Nov 27 2022 02:09:33
%S A323921 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,
%T A323921 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,
%U A323921 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
%N A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.
%C A323921 Sum of powers of 4 dividing n.
%H A323921 Amiram Eldar, <a href="/A323921/b323921.txt">Table of n, a(n) for n = 1..10000</a>
%F A323921 G.f.: Sum_{k>=0} 4^k * x^(4^k) / (1 - x^(4^k)).
%F A323921 L.g.f.: -log(Product_{k>=0} (1 - x^(4^k))).
%F A323921 Dirichlet g.f.: zeta(s) / (1 - 4^(1 - s)).
%F A323921 From _Amiram Eldar_, Nov 27 2022: (Start)
%F A323921 Multiplicative with a(2^e) = (4^floor((e+2)/2)-1)/3, and a(p^e) = 1 for p != 2.
%F A323921 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)
%t A323921 Table[(4^(IntegerExponent[n, 4] + 1) - 1)/3, {n, 1, 100}]
%t A323921 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
%o A323921 (Python)
%o A323921 def A323921(n): return ((1<<((~n&n-1).bit_length()&-2)+2)-1)//3 # _Chai Wah Wu_, Jul 09 2022
%o A323921 (PARI) a(n) = (4^(valuation(n, 4) + 1) - 1) / 3; \\ _Michel Marcus_, Jul 09 2022
%Y A323921 Cf. A001620, A038712, A080278, A088842, A234957, A235127, A339747, A339748.
%K A323921 nonn,mult
%O A323921 1,4
%A A323921 _Ilya Gutkovskiy_, Dec 15 2020