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A263825
Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.
7
1, 7, 5, 23, 7, 39, 9, 65, 18, 61, 13, 143, 15, 87, 35, 183, 19, 182, 21, 245, 45, 151, 25, 465, 38, 189, 58, 375, 31, 429, 33, 549, 65, 277, 63, 806, 39, 327, 75, 875, 43, 663, 45, 719, 126, 439, 49, 1535, 66, 650, 95, 933, 55, 982, 91, 1425, 105, 637, 61, 2093
OFFSET
1,2
LINKS
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:
a := a+A059376(l)*s1 + s2*s3 + A007434(l)*s4 ;
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
];
Array[A263825, 60] (* Jean-François Alcover, Nov 21 2017, after R. J. Mathar *)
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));
S11(n) = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n) = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = { 1/n * sumdiv(n, d,
A059376(d) * S1(n\d) + EpiPcZn(d) * S11(n\d) + A007434(d) * S21(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