A263830 -id:A263830 - cp's OEIS frontend

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

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825-A263830, A263832.

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

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 *)

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

A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

Original entry on oeis.org

1, 0, 5, 2, 7, 0, 9, 6, 18, 0, 13, 10, 15, 0, 35, 14, 19, 0, 21, 14, 45, 0, 25, 30, 38, 0, 58, 18, 31, 0, 33, 30, 65, 0, 63, 36, 39, 0, 75, 42, 43, 0, 45, 26, 126, 0, 49, 70, 66, 0, 95, 30, 55, 0, 91, 54, 105, 0, 61, 70, 63, 0, 162, 62, 105, 0, 69

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825, A263826, A263827, A263828, A263829, A263830, A263831.

sigma[n_] := DivisorSigma[1, n]; q = Quotient;
a[n_] := Switch[Mod[n, 4], 0, Sum[sigma[q[n, 2d]] - sigma[q[n, 4d]], {d, Divisors[q[n, 4]]}], 2, 0, 1|3, Sum[sigma[d], {d, Divisors[n]}]];
Array[a, 70] (* Jean-François Alcover, Dec 01 2018, after Gheorghe Coserea *)

A007429(n) = sumdiv(n, d, sigma(d));
a(n) = {
  if (n%2, A007429(n), if (n%4, 0,
      sumdiv(n\4, d, sigma(n\(2*d)) - sigma(n\(4*d)))));
};
vector(67, n, a(n))  \\ Gheorghe Coserea, May 05 2016

More terms from Gheorghe Coserea, May 05 2016

A263826 The number c_{Z3 pi_1(B_1)}(2n) of 3-torus 2n-coverings over the first amphicosm.

Original entry on oeis.org

1, 7, 9, 29, 19, 63, 33, 107, 74, 133, 73, 285, 99, 231, 219, 393, 163, 518, 201, 623, 393, 511, 289, 1155, 422, 693, 634, 1101, 451, 1533, 513, 1479, 897, 1141, 915, 2482, 723, 1407, 1227, 2609, 883, 2751, 969, 2477, 2078, 2023, 1153, 4569

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825-A263830, A263832.

a[n_] := 1/2 Sum[Sum[(d^2 + 5/2 + 3/2 (-1)^Mod[d, 2]) m, {m, Divisors[n/d]} ], {d, Divisors[n]}];
Array[a, 48] (* Jean-François Alcover, Oct 10 2018, after Gheorghe Coserea *)

a(n) = 1/2 * sumdiv(n, d, sumdiv(n\d, m, (d^2 + 5/2 + 3/2*(-1)^(d%2))*m));
vector(48, n, a(n))  \\ Gheorghe Coserea, May 04 2016

More terms from Gheorghe Coserea, May 04 2016

A263827 The number c_{Cc pi_1(B_1)}(2n) of the second amphicosm 2n-coverings over the first amphicosm.

Original entry on oeis.org

2, 6, 10, 14, 14, 30, 18, 30, 36, 42, 26, 70, 30, 54, 70, 62, 38, 108, 42, 98, 90, 78, 50, 150, 76, 90, 116, 126, 62, 210, 66, 126, 130, 114, 126, 252, 78, 126, 150, 210, 86, 270, 90, 182, 252, 150, 98, 310, 132, 228, 190, 210, 110, 348, 182, 270, 210, 186, 122, 490, 126, 198, 324, 254, 210, 390, 138, 266, 250, 378

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825-A263830, A263832.

A263827 := proc(n)
    local locn,a,twol,fourl ;
    locn := 2*n ;
    # Theorem 3 (iii)
    a := 0 ;
    for twol in numtheory[divisors](locn) do
        if type(twol,'even') then
            a := a+numtheory[sigma](locn/twol) ;
        end if;
    end do:
    for fourl in numtheory[divisors](locn) do
        if modp(fourl,4) = 0 then
            a := a-numtheory[sigma](locn/fourl) ;
        end if;
    end do:
    %*2 ;
end proc: # R. J. Mathar, Nov 03 2015

a[n_] := 2*Sum[If[Mod[d,4] == 2, DivisorSigma[1, 2*n/d], 0], {d, Divisors[ 2*n ] } ];
Array[a, 70] (* Jean-François Alcover, Dec 03 2017 *)

A007429(n) = sumdiv(n, d, sigma(d));
a(n) = 2*A007429(n) - if(n%2, 0, 2*A007429(n\2));
vector(70, n, a(n))  \\ Gheorghe Coserea, May 04 2016

A263828 The number c_{P c pi_1(B_1)}(n) of the first amphicosm n-coverings over the first amphicosm.

Original entry on oeis.org

1, 4, 5, 10, 7, 20, 9, 22, 18, 28, 13, 50, 15, 36, 35, 46, 19, 72, 21, 70, 45, 52, 25, 110, 38, 60, 58, 90, 31, 140, 33, 94, 65, 76, 63, 180, 39, 84, 75, 154, 43, 180, 45, 130, 126, 100, 49, 230, 66, 152, 95, 150, 55, 232, 91, 198, 105, 124, 61

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825-A263830, A263832.

a[n_] := Sum[(3/2 + 1/2 (-1)^Mod[d, 2]) DivisorSigma[1, n/d], {d, Divisors[ n]}] - If[OddQ[n], 0, Sum[(3/2 + 1/2 (-1)^Mod[d, 2]) DivisorSigma[1, n/(2 d)], {d, Divisors[n/2]}]];
Array[a, 59] (* Jean-François Alcover, Oct 10 2018, after Gheorghe Coserea *)

a(n) = {
  sumdiv(n, d, (3/2 + 1/2*(-1)^(d%2)) * sigma(n/d)) -
  if (n%2, 0, sumdiv(n\2, d, (3/2 + 1/2*(-1)^(d%2))*sigma(n\(2*d))))
};
vector(59, n, a(n))  \\ Gheorghe Coserea, May 04 2016

More terms from Gheorghe Coserea, May 04 2016

A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

Original entry on oeis.org

1, 3, 5, 13, 7, 19, 9, 43, 18, 33, 13, 93, 15, 51, 35, 137, 19, 110, 21, 175, 45, 99, 25, 355, 38, 129, 58, 285, 31, 289, 33, 455, 65, 201, 63, 626, 39, 243, 75, 721, 43, 483, 45, 589, 126, 339, 49, 1305, 66, 498, 95, 783, 55, 750, 91, 1227

Offset: 1

N. J. A. Sloane, Oct 28 2015

nonn

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.

Cf. A263825-A263830, A263832.

A001001(n) = sumdiv(n, d, sigma(d) * d);
A007429(n) = sumdiv(n, d, sigma(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));
S22(n)     = { if (n%2, A060640(n), if (n%4, 0,
  sumdiv(n\4, d, 2*d*(sigma(n\(2*d)) - sigma(n\(4*d))))));
};
A027844(n) = S1(n) + S11(n) + S21(n);
a(n) = { 1/n * sumdiv(n, d,
  A059376(d) * S1(n\d) + EpiPcZn(d) * S21(n\d) + A007434(d) * S22(n\d));
};
vector(56, n, a(n))  \\ Gheorghe Coserea, May 04 2016

More terms from Gheorghe Coserea, May 04 2016

cp's OEIS Frontend

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

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A263832 The number c_{Cc,pi_1(B_2)}(n) of the second amphicosm n-coverings over the second amphicosm.

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A263826 The number c_{Z3 pi_1(B_1)}(2n) of 3-torus 2n-coverings over the first amphicosm.

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A263827 The number c_{Cc pi_1(B_1)}(2n) of the second amphicosm 2n-coverings over the first amphicosm.

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A263828 The number c_{P c pi_1(B_1)}(n) of the first amphicosm n-coverings over the first amphicosm.

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A263829 Total number c_{pi_1(B_2)}(n) of n-coverings over the second amphicosm.

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