User:Jason Kimberley/lattice numerology

A218147

Empirical data

From BBCZ , 2012/10/11:

$d$	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
$n(d)$	2	2	4	4	12	8	18	8	30	16	36	24	32	32	64	36	90	32	96	60	132	64	100	72		96		64		128

PSLQ recovered a degree $n(d)$ polynomial satisfied by

${\rm {exp}}\left(8\pi \phi _{2}\left({\frac {1}{d}},{\frac {1}{d}}\right)\right)$,

where

$\phi _{2}(r_{1},r_{2})={\frac {1}{\pi ^{2}}}\sum _{m_{1},m_{2}\,\in \,\mathbb {Z} \backslash 2\mathbb {Z} }{\frac {\cos(\pi m_{1}r_{1})\cos(\pi m_{2}r_{2})}{m_{1}^{2}+m_{2}^{2}}}.$

Hypothesis

• For prime $p$, define $n(p)$ by

$n(2)={\frac {1}{2}},\;$

$n(4k+1)=(2k)^{2},\;$

$n(4k-1)=2k\,(2k-1);\;$

• For prime $p$ and any $a\in \mathbb {N}$,

$n(p^{2}a)=p^{2}n(pa);$

• For coprime $a,b$

$n(ab)\,=\,4\,n(a)\,n(b).$

I have defined A218147 as this sequence.

Plots

(d, n)

(d^2, n)

(log d, log n)

Consequent properties

• $n(d)$ is not a multiplicative sequence; however, it is a divisibility sequence.

• $4\,n(d)\leq d^{2}$

• If we let

$d=\prod _{i=1}^{k}{p_{i}^{e_{i}}},$

then

$n\left(d\right)=4^{k-1}\,{\prod _{i=1}^{k}{p_{i}^{2e_{i}-2}\,n(p_{i})}}.$

Since

${\prod _{i=1}^{k}p_{i}^{e_{i}-1}}={\frac {d}{{\rm {A007947}}(d)}}={\rm {A003557}}(d),$

where A007947 is the "squarefree kernel" sequence and A003557 is the quotient,

$n(d)=4^{k-1}{\rm {A003557}}(d)^{2}\,\prod _{i=1}^{k}{n(p_{i})}.$

Magma input

 n :=
   func<
     d
   |
       d eq 2
     select
       1/2
     else
         IsPrime(d)
       select
           d mod 4 eq 1
         select
           (d div 2)^2
         else // p mod 4 eq 3
           (d div 2)*(d div 2 + 1)
       else
         4^(#fact-1) *
         &*[
           Rationals()
         |
           p^(2*e-2) *
           $$(p) // recursion
           where p, e
           is Explode(p_e)
         :
           p_e in fact
         ]
         where
           fact
         is
           Factorisation(d)
   >;
 
 [<d,n(d)>:d in[1..64]];

Paste that into the Magma Calculator.

Magma output

 [ <1, 1/4>, <2, 1/2>, <3, 2>, <4, 2>, <5, 4>, <6, 4>, <7, 12>, <8, 8>, <9, 18>,
   <10, 8>, <11, 30>, <12, 16>, <13, 36>, <14, 24>, <15, 32>, <16, 32>, <17, 64>,
   <18, 36>, <19, 90>, <20, 32>, <21, 96>, <22, 60>, <23, 132>, <24, 64>, <25,100>,
   <26, 72>, <27, 162>, <28, 96>, <29, 196>, <30, 64>, <31, 240>, <32, 128>,
   <33, 240>, <34, 128>, <35, 192>, <36, 144>, <37, 324>, <38, 180>, <39, 288>,
   <40, 128>, <41, 400>, <42, 192>, <43, 462>, <44, 240>, <45, 288>, <46, 264>,
   <47, 552>, <48, 256>, <49, 588>, <50, 200>, <51, 512>, <52, 288>, <53, 676>,
   <54, 324>, <55, 480>, <56, 384>, <57, 720>, <58, 392>, <59, 870>, <60, 256>,
   <61, 900>, <62, 480>, <63, 864>, <64, 512> ]

Further empirical data

My conjectured values of $n(27)=162,\ n(29)=196\,$ and $\,n(31)=240$ are confirmed. [Verbal communication from Jon Borwein 2012/11/23.]

Questions

• Do $n(1)=1/4$, and $n(2)=1/2$ have meaning in the original problem?

---

Compressed Poisson

Empirical Data

$d$	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32
$m(d)$		4	4	8	3	16	6	16	15	16	12	49	8	32	32	24	27	64	24	40	33	64	20		27	48	85	64	45

 m := [0, 0, 0, 4, 4, 8, 3, 16, 6, 16, 15, 16, 12, 49, 8, 32, 32, 24, 27, 64, 24, 40, 33, 64, 20, 0, 27, 48, 85, 64, 45, 0];

Observations

• $m(p)$ is a multiple of ${\frac {p-1}{2}}$ for each odd prime $p\neq 29$
  • $m(29)=85=6(29-1)/2+1$
  • $m(p)={\frac {3(p-1)}{2}}$ for $7\neq p\equiv 3\!\!\!\!{\pmod {4}}$
• $m$ is not a divisibility sequence:
  • $7\mid 14$ but $m(7)=3\nmid 49=m(14)$
  • $9\mid 27$ but $m(9)=6\nmid 27=m(27)$
  • $11\mid 22$ but $m(11)=15\nmid 40=m(22)$
  • $14\mid 28$ but $m(14)=49\nmid 48=m(28)$

Further and Corrected Empirical Data

From BB

$d$	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40
$m(d)$	1	4	4	8	3	16	6	16	15	16	12	48	8	32	32	24	27	64	24	40	33	64	20		27	48	84	64	45		40		72		36		48

 m := [0, 0, 1, 4, 4, 8, 3, 16, 6, 16, 15, 16, 12, 48, 8, 32, 32, 24, 27, 64, 24, 40, 33, 64, 20, 0, 27, 48, 85, 64, 45, 0, 40, 0, 72, 36, 0, 48, 0];

Updated Observations

• $m(p)$ is a multiple of ${\frac {p-1}{2}}$ for each odd prime.
  • $m(p)=p-1$ for $p\equiv 1\!\!\!\!{\pmod {4}}$ other than:
    • $m(17)=2(17-1)$
    • $m(29)=3(29-1)$
  • $m(p)={\frac {3(p-1)}{2}}$ for $p\equiv 3\!\!\!\!{\pmod {4}}$ other than:
    • $m(3)={\frac {3-1}{2}}$
    • $m(7)={\frac {7-1}{2}}$
• $m$ is not a divisibility sequence:
  • $9\mid 27$ but $m(9)=6\nmid 27=m(27)$
  • $11\mid 22$ but $m(11)=15\nmid 40=m(22)$
  • $11\mid 33$ but $m(11)=15\nmid 40=m(33)$
• From J. M. Borwein: $2\phi (d){\mid }m(d)$
• A000010(d) = A003557(d) * A173557(d). Cf. n(d) as a multiple of A003557 above.
• A000010(d)/2 = A023022(d), for d > 2, is the degree of the minimal polynomial of $\cos(2\pi /d)$.