cp's OEIS Frontend

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

%I A263825 #24 Nov 22 2017 03:20:13
%S A263825 1,7,5,23,7,39,9,65,18,61,13,143,15,87,35,183,19,182,21,245,45,151,25,
%T A263825 465,38,189,58,375,31,429,33,549,65,277,63,806,39,327,75,875,43,663,
%U A263825 45,719,126,439,49,1535,66,650,95,933,55,982,91,1425,105,637,61,2093
%N A263825 Total number c_{pi_1(B_1)}(n) of n-coverings over the first amphicosm.
%H A263825 Gheorghe Coserea, <a href="/A263825/b263825.txt">Table of n, a(n) for n = 1..20000</a>
%H A263825 G. Chelnokov, M. Deryagina, A. Mednykh, <a href="http://arxiv.org/abs/1502.01528">On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2</a>, arXiv preprint arXiv:1502.01528 [math.AT], 2015.
%H A263825 G. Chelnokov, M. Deryagina and A. Mednykh, <a href="https://doi.org/10.1080/00927872.2016.1222396">On the coverings of Euclidean manifolds B_1 and B_2</a>, Communications in Algebra, Vol. 45, No. 4 (2017), 1558-1576.
%p A263825 A263825 := proc(n)
%p A263825     local a,l,m,s1,s2,s3,s4 ;
%p A263825     # Theorem 2
%p A263825     a := 0 ;
%p A263825     for l in numtheory[divisors](n) do
%p A263825         m := n/l ;
%p A263825         s1 := 0 ;
%p A263825         for twok in numtheory[divisors](m) do
%p A263825             if type(twok,'even') then
%p A263825                 k := twok/2 ;
%p A263825                 s1 := s1+numtheory[sigma](k)*k ;
%p A263825             end if;
%p A263825         end do:
%p A263825         s2 := 0 ;
%p A263825         for d in numtheory[divisors](l) do
%p A263825             s2 := s2+numtheory[mobius](l/d)*d^2*igcd(2,d) ;
%p A263825         end do:
%p A263825         s3 := 0 ;
%p A263825         for k in numtheory[divisors](m) do
%p A263825             s3 := s3+numtheory[sigma](m/k)*k ;
%p A263825             if modp(m,2*k) = 0 then
%p A263825                 s3 := s3-numtheory[sigma](m/2/k)*k ;
%p A263825             end if;
%p A263825         end do:
%p A263825         s4 := 0 ;
%p A263825         for twok in numtheory[divisors](m) do
%p A263825             if type(twok,'even') then
%p A263825                 s4 := s4+numtheory[sigma](m/twok)*twok ;
%p A263825                 if modp(m,2*twok) = 0 then
%p A263825                     s4 := s4-numtheory[sigma](m/2/twok)*twok ;
%p A263825                 end if;
%p A263825             end if;
%p A263825         end do:
%p A263825         a := a+A059376(l)*s1 + s2*s3 + A007434(l)*s4 ;
%p A263825     end do:
%p A263825     a/n ;
%p A263825 end proc: # _R. J. Mathar_, Nov 03 2015
%t A263825 A007434[n_] := Sum[ MoebiusMu[n/d] * d^2, {d, Divisors[n]}];
%t A263825 A059376[n_] := Sum[ MoebiusMu[n/d] * d^3, {d, Divisors[n]}];
%t A263825 A263825[n_] := Module[{a, l, m, s1, s2, s3, s4},
%t A263825 a = 0;
%t A263825 Do[m = n/l;
%t A263825 s1 = 0; Do[If[EvenQ[twok], k = twok/2; s1 = s1 + DivisorSigma[1, k]*k], {twok, Divisors[m]}];
%t A263825 s2 = 0; Do[s2 = s2 + MoebiusMu[l/d]*d^2*GCD[2, d], {d, Divisors[l]}];
%t A263825 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]}];
%t A263825 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,
%t A263825 {l, Divisors[n]}]; a/n
%t A263825 ];
%t A263825 Array[A263825, 60] (* _Jean-François Alcover_, Nov 21 2017, after _R. J. Mathar_ *)
%o A263825 (PARI)
%o A263825 A001001(n) = sumdiv(n, d, sigma(d) * d);
%o A263825 A007434(n) = sumdiv(n, d, moebius(n\d) * d^2);
%o A263825 A059376(n) = sumdiv(n, d, moebius(n\d) * d^3);
%o A263825 A060640(n) = sumdiv(n, d, sigma(n\d) * d);
%o A263825 EpiPcZn(n) = sumdiv(n, d, moebius(n\d) * d^2 * gcd(d,2));
%o A263825 S1(n)      = if (n%2, 0, A001001(n\2));
%o A263825 S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
%o A263825 S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
%o A263825 a(n) = { 1/n * sumdiv(n, d,
%o A263825   A059376(d) * S1(n\d) + EpiPcZn(d) * S11(n\d) + A007434(d) * S21(n\d));
%o A263825 };
%o A263825 vector(60, n, a(n))  \\ _Gheorghe Coserea_, May 04 2016
%Y A263825 Cf. A263825-A263830, A263832.
%K A263825 nonn
%O A263825 1,2
%A A263825 _N. J. A. Sloane_, Oct 28 2015