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User:Jason Kimberley/lattice numerology
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A218147
Empirical data
From BBCZ [1], 2012/10/11:
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 | |
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 polynomial satisfied by
- ,
where
Hypothesis
- For prime , define by
- For prime and any ,
- For coprime
I have defined A218147 as this sequence.
Plots
Consequent properties
- is not a multiplicative sequence; however, it is a divisibility sequence.
- If we let
then
Since
where A007947 is the "squarefree kernel" sequence and A003557 is the quotient,
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 and are confirmed. [Verbal communication from Jon Borwein 2012/11/23.]
Questions
- Do , and have meaning in the original problem?
Compressed Poisson
Empirical Data
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 | |
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
- is a multiple of for each odd prime
- for
- is not a divisibility sequence:
- but
- but
- but
- but
Further and Corrected Empirical Data
From BB [2]
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 | |
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
- is a multiple of for each odd prime.
- for other than:
- for other than:
- for other than:
- is not a divisibility sequence:
- but
- but
- but
- From J. M. Borwein:
- 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 .
References
- ↑ D. H. Bailey, J. Borwein, R. Crandall and J. Zucker (2012), Lattice sums arising from the Poisson equation, preprint.
- ↑ D. H. Bailey and J. M. Borwein (2012), Compressed lattice sums arising from the Poisson equation, preprint.