A360156 a(n) is the sum of the even unitary divisors of 2*n.

2, 4, 8, 8, 12, 16, 16, 16, 20, 24, 24, 32, 28, 32, 48, 32, 36, 40, 40, 48, 64, 48, 48, 64, 52, 56, 56, 64, 60, 96, 64, 64, 96, 72, 96, 80, 76, 80, 112, 96, 84, 128, 88, 96, 120, 96, 96, 128, 100, 104, 144, 112, 108, 112, 144, 128, 160, 120, 120, 192, 124, 128

Offset: 1

Author

Amiram Eldar, Jan 28 2023

Keywords

• nonn
• easy

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Octavio A. Agustín-Aquino, Wang-Sun Formula in GL(Z/2kZ), INTEGERS, Vol. 23 (2023), #A37; arXiv preprint, arXiv:2207.14495 [math.NT], 2022.

Crossrefs

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

Programs

Mathematica

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]

PARI

usigma(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + 1)} ;
a(n) = {my(e = valuation(n, 2)); (1 << (e+1)) * usigma(n >> e); }

Formula

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

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

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

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

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

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.