cp's OEIS Frontend

A218147 Degree of minimal polynomial satisfied by exp(8*Pi*phi_2(1/n,1/n)), where phi_2 is defined in the Comments.

%I A218147 #109 Apr 06 2024 09:03:11
%S A218147 2,2,4,4,12,8,18,8,30,16,36,24,32,32,64,36,90,32,96,60,132,64,100,72,
%T A218147 162,96,196,64,240,128,240,128,192,144,324,180,288,128,400,192,462,
%U A218147 240,288,264,552,256,588,200,512,288,676,324,480,384,720,392,870,256
%N A218147 Degree of minimal polynomial satisfied by exp(8*Pi*phi_2(1/n,1/n)), where phi_2 is defined in the Comments.
%C A218147 Crandall defines phi_2(r_1,r_2) = (1/Pi^2) Sum_{positive & negative odd m_1, m_2} cos(Pi m_1 r_1) cos(Pi m_2 r_2) / (m_1^2+m_2^2).
%C A218147 Lemma: 4a(n) < n^2. Proof: 4a(2) = 2 < 2^2; 4a(4k+1) = 16k^2 < (4k+1)^2; 4a(4k+3) = (4k+2)(4k+4) = (4k+3)^2-1; 4a(p^2 k) = 4p^2 a(pk) < p^2(pk)^2 = (p^2 k)^2; 4 a(jk) = 4 a(j) 4 a(k) < (jk)^2.
%C A218147 Corollary: a(n) <=  A198442(n).
%D A218147 R. Crandall, The Poisson equation and "natural" Madelung constants, preprint 2012 (see section 2 of BBCZ below).
%H A218147 Jason Kimberley, <a href="/A218147/b218147.txt">Table of n, a(n) for n = 3..10000</a>
%H A218147 D. H. Bailey, J. Borwein, R. Crandall and J. Zucker, <a href="http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/PoissonLattice.pdf">Lattice sums arising from the Poisson equation</a>, preprint (2012).
%H A218147 D. H. Bailey, J. M. Borwein and J. S. Kimberley, with an appendix by W. B. Ladd, <a href="http://www.davidhbailey.com/dhbpapers/poisson-res.pdf">Computer discovery and analysis of large Poisson polynomials</a>, Experimental Mathematics, August 2016.
%H A218147 J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H A218147 J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H A218147 OEIS (Plot 2), <a href="/plot2a?name1=A000027&amp;name2=A218147&amp;tform1=log+base+10&amp;tform2=log+base+10&amp;shift=0&amp;radiop1=xy&amp;drawpoints=true">Plot of (log n, log a(n))</a>
%H A218147 OEIS (Plot 2), <a href="https://oeis.org/plot2a?name1=A218147&amp;name2=A198442&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=matp&amp;drawpoints=true">Plot of (n, a(n)) and (n, A198442(n))</a>
%F A218147 a(n) = A079458(n) / 4, for n > 2. - _Jason Kimberley_, Nov 14 2015
%F A218147 Watson Ladd has proved that the sequence satisfies the following recurrence relations, which were conjectured by _Jason Kimberley_:
%F A218147 a(1) = 1/4, a(2) = 1/2, for notational convenience;
%F A218147 a(4k+1) = (2k)*(2k) for prime 4k+1;
%F A218147 a(4k+3) = (2k+1)*(2k+2) for prime 4k+3;
%F A218147 a(p^2 k) = p^2 * a(p*k) for prime p;
%F A218147 a(jk) = 4*a(j)*a(k) for j coprime to k.
%o A218147 (Magma) A218147 := func<n|n eq 2 select 1/2 else IsPrime(n)select
%o A218147 n mod 4 eq 1 select(n div 2)^2 else(n div 2)*(n div 2+1)
%o A218147 else(4^(#fact-1)*&*[p^(2*e-2)*$$(p)where p,e is Explode(p_e):p_e in fact]
%o A218147 where fact is Factorisation(n))>; // _Jason Kimberley_, Oct 23 2012
%o A218147 (Magma) A218147 := func<n|#UnitGroup(quo<IntegerRing(QuadraticField(-1))|n>)/4>; // _Jason Kimberley_, Nov 14 2015
%Y A218147 Cf. A079458, A198442.
%K A218147 nonn,easy,nice
%O A218147 3,1
%A A218147 _Jason Kimberley_, Oct 21 2012 and Apr 04 2016
%E A218147 Entry revised by _N. J. A. Sloane_, May 15 2016, to take into account the fact that the conjectured formula for this sequence has now been established by Watson Ladd.