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A360156
a(n) is the sum of the even unitary divisors of 2*n.
2
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
1,1
COMMENTS
a(n) is the unitary analog of A146076(2*n).
LINKS
Octavio A. Agustín-Aquino, Wang-Sun Formula in GL(Z/2kZ), INTEGERS, Vol. 23 (2023), #A37; arXiv preprint, arXiv:2207.14495 [math.NT], 2022.
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.
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]
PROG
(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); }
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 28 2023
STATUS
approved