A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.

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

Author

N. J. A. Sloane, Oct 28 2015

Keywords

• nonn

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.

Crossrefs

Cf. A263825-A263830, A263832.

Programs

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 *)

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