A360156 - OEIS

%I #29 Jun 08 2023 07:13:46

%S 2,4,8,8,12,16,16,16,20,24,24,32,28,32,48,32,36,40,40,48,64,48,48,64,

%T 52,56,56,64,60,96,64,64,96,72,96,80,76,80,112,96,84,128,88,96,120,96,

%U 96,128,100,104,144,112,108,112,144,128,160,120,120,192,124,128

%N a(n) is the sum of the even unitary divisors of 2*n.

%C a(n) is the unitary analog of A146076(2*n).

%H Amiram Eldar, <a href="/A360156/b360156.txt">Table of n, a(n) for n = 1..10000</a>

%H Octavio A. Agustín-Aquino, <a href="http://math.colgate.edu/~integers/x37/x37.pdf">Wang-Sun Formula in GL(Z/2kZ)</a>, INTEGERS, Vol. 23 (2023), #A37; <a href="https://arxiv.org/abs/2207.14495">arXiv preprint</a>, arXiv:2207.14495 [math.NT], 2022.

%F a(n) = Sum_{even d|(2*n), gcd(d, 2*n/d)=1} d.

%F a(n) = A034448(2*n) - A192066(2*n).

%F a(n) = A192066(2*n) - A328258(2*n).

%F a(n) = A171977(n) * A192066(n).

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (7*zeta(3)).

%F Dirichlet g.f. of b(n): (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(s+1)-2)/(2^(2*s)-2), where b(n) is the sum of the even unitary divisors of n: b(n) = a(n/2) if n is even and 0 otherwise.

%t usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^(e + 1) * usigma[n/2^e]]; Array[a, 100]

%o (PARI) usigma(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + 1)} ;

%o a(n) = {my(e = valuation(n, 2)); (1 << (e+1)) * usigma(n >> e); }

%Y Cf. A002117, A034448, A100008, A146076, A171977, A192066, A328258.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Jan 28 2023