OFFSET
1,2
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..20000
G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
G. Chelnokov, M. Deryagina and A. Mednykh, On the coverings of Euclidean manifolds B_1 and B_2, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.
MAPLE
A263825 := proc(n)
local a, l, m, s1, s2, s3, s4 ;
# Theorem 2
a := 0 ;
for l in numtheory[divisors](n) do
m := n/l ;
s1 := 0 ;
for twok in numtheory[divisors](m) do
if type(twok, 'even') then
k := twok/2 ;
s1 := s1+numtheory[sigma](k)*k ;
end if;
end do:
s2 := 0 ;
for d in numtheory[divisors](l) do
s2 := s2+numtheory[mobius](l/d)*d^2*igcd(2, d) ;
end do:
s3 := 0 ;
for k in numtheory[divisors](m) do
s3 := s3+numtheory[sigma](m/k)*k ;
if modp(m, 2*k) = 0 then
s3 := s3-numtheory[sigma](m/2/k)*k ;
end if;
end do:
s4 := 0 ;
for twok in numtheory[divisors](m) do
if type(twok, 'even') then
s4 := s4+numtheory[sigma](m/twok)*twok ;
if modp(m, 2*twok) = 0 then
s4 := s4-numtheory[sigma](m/2/twok)*twok ;
end if;
end if;
end do:
end do:
a/n ;
end proc: # R. J. Mathar, Nov 03 2015
MATHEMATICA
A007434[n_] := Sum[ MoebiusMu[n/d] * d^2, {d, Divisors[n]}];
A059376[n_] := Sum[ MoebiusMu[n/d] * d^3, {d, Divisors[n]}];
A263825[n_] := Module[{a, l, m, s1, s2, s3, s4},
a = 0;
Do[m = n/l;
s1 = 0; Do[If[EvenQ[twok], k = twok/2; s1 = s1 + DivisorSigma[1, k]*k], {twok, Divisors[m]}];
s2 = 0; Do[s2 = s2 + MoebiusMu[l/d]*d^2*GCD[2, d], {d, Divisors[l]}];
s3 = 0; Do[s3 = s3 + DivisorSigma[1, m/k]*k ; If[Mod[m, 2*k] == 0, s3 = s3 - DivisorSigma[1, m/2/k]*k], {k, Divisors[m]}];
s4 = 0; Do[If[EvenQ[twok], s4 = s4 + DivisorSigma[1, m/twok]*twok; If[ Mod[m, 2*twok] == 0, s4 = s4 - DivisorSigma[1, m/2/twok]*twok]], {twok, Divisors[m]}]; a = a + A059376[l]*s1 + s2*s3 + A007434[l]*s4,
{l, Divisors[n]}]; a/n
];
PROG
(PARI)
A001001(n) = sumdiv(n, d, sigma(d) * d);
A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d, 2));
S1(n) = if (n%2, 0, A001001(n\2));
a(n) = { 1/n * sumdiv(n, d,
};
vector(60, n, a(n)) \\ Gheorghe Coserea, May 04 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 28 2015
STATUS
approved