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

2, 2, 4, 4, 12, 8, 18, 8, 30, 16, 36, 24, 32, 32, 64, 36, 90, 32, 96, 60, 132, 64, 100, 72, 162, 96, 196, 64, 240, 128, 240, 128, 192, 144, 324, 180, 288, 128, 400, 192, 462, 240, 288, 264, 552, 256, 588, 200, 512, 288, 676, 324, 480, 384, 720, 392, 870, 256

Offset

3,1

Comments

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).

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.

Corollary: a(n) <= A198442(n).

References

R. Crandall, The Poisson equation and "natural" Madelung constants, preprint 2012 (see section 2 of BBCZ below).

Links

Jason Kimberley, Table of n, a(n) for n = 3..10000

D. H. Bailey, J. Borwein, R. Crandall and J. Zucker, Lattice sums arising from the Poisson equation, preprint (2012).

D. H. Bailey, J. M. Borwein and J. S. Kimberley, with an appendix by W. B. Ladd, Computer discovery and analysis of large Poisson polynomials, Experimental Mathematics, August 2016.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

OEIS (Plot 2), Plot of (log n, log a(n))

OEIS (Plot 2), Plot of (n, a(n)) and (n, A198442(n))

Formula

a(n) = A079458(n) / 4, for n > 2. - Jason Kimberley, Nov 14 2015

Watson Ladd has proved that the sequence satisfies the following recurrence relations, which were conjectured by Jason Kimberley:

a(1) = 1/4, a(2) = 1/2, for notational convenience;

a(4k+1) = (2k)*(2k) for prime 4k+1;

a(4k+3) = (2k+1)*(2k+2) for prime 4k+3;

a(p^2 k) = p^2 * a(p*k) for prime p;

a(jk) = 4*a(j)*a(k) for j coprime to k.

Prog

(Magma) A218147 := func<n|n eq 2 select 1/2 else IsPrime(n)select
n mod 4 eq 1 select(n div 2)^2 else(n div 2)*(n div 2+1)
else(4^(#fact-1)*&*[p^(2*e-2)*$$(p)where p, e is Explode(p_e):p_e in fact]
where fact is Factorisation(n))>; // Jason Kimberley, Oct 23 2012
(Magma) A218147 := func<n|#UnitGroup(quo<IntegerRing(QuadraticField(-1))|n>)/4>; // Jason Kimberley, Nov 14 2015

Crossrefs

Cf. A079458, A198442.

Sequence in context: A288044 A081164 A125553 * A226978 A243331 A300218

Adjacent sequences: A218144 A218145 A218146 * A218148 A218149 A218150

Keyword

nonn,easy,nice,changed

Author

Jason Kimberley, Oct 21 2012 and Apr 04 2016

Extensions

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.

Status

approved