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MathGroup Archive 2007

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Re: Re: request for a few minutes CPU-time

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79814] Re: [mg79521] Re: [mg79477] request for a few minutes CPU-time
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sun, 5 Aug 2007 04:55:21 -0400 (EDT)
  • References: <200707271001.GAA03340@smc.vnet.net> <200707280937.FAA00004@smc.vnet.net>

Daniel Lichtblau wrote:
> Peter Pein wrote:
> 
>>Dear group,
>>
>>I've written code, which looks for numbers for which a smaller natural
>>number exists which has the same sum of cubes of it's divisors:
>>
>>Block[{spa, nmax = 6*106, expo = 3},
>>   Reap[For[n = 1, n <= nmax, n++,
>>      (If[Head[#1] === spa, #1 = n, Sow[{n, #1}]] & )[
>>       spa[DivisorSigma[expo, n]]]]][[2,1]]]
>>
>>the name spa is an artefact; I tried this with SparseArrays, but the
>>allowed range of indices has not been sufficient. I now use spa as an
>>initially undefined function (Block[{spa..}]). For each n to test I look
>>wether spa[sigma(r,n)] has been defined. If not, the Head is still spa
>>and I set spa[sigma(r,n)] to n; else the remembered value together with
>>n will go to the result via the Sow-Reap mechanism.
>>
>> If you've got RAM (4GB or so) than I do (1.5 GB),
>>could you please run this code  with, say nmax=10 or 20 million? On my
>>machine it swapped heavily with nmax=6 million and I had to kill
>>MathKernel as I tried nmax=10^7. The lines above took ~181 seconds to
>>evaluate (nmax=10^7 has been stopped by me after 15 minutes). I do not
>>expect any runtimes of more than ~7-10 minutes. Would this be possible,
>>please?
>>
>> Alternatively any hints how to calculate these sequences more efficient
>>would be highly appreciated (AFAIK there exists no kind of "inverse
>>function" to sigma(r,n) w.r.t. n which could be calculated without this
>>brute-force method).
>>
>>Thank you for your attention and in advance for CPU-time,
>>
>>Peter
> 

Here is a way that removes excess storage, requiring only 8-10% or so of 
the previous memoization-based code I had used. The idea is to discard 
down values once they are known not to be needed.

duplicatedDivisorSumCubes[n_] := Module[
   {c, dsc, cdsc, jlo = 1, cinv, len = Ceiling[.064*n]},
   Off[Unset::"norep"];
   cinv = Table[0, {len}];
   Reap[Do[
     dsc = DivisorSigma[3, j];
     cdsc = c[dsc];
     cinv[[Mod[j, len, 1]]] = dsc;
     If[IntegerQ[cdsc],
       Sow[{j, cdsc, dsc}]];
     c[dsc] = j;
     While[jlo*1.064 < j, c[cinv[[Mod[jlo, len, 1]]]] =.; jlo++],
     {j, 1, n}]][[2, 1]]
   ]

The Off[...] is due to the fact that c[j] is not quite a 1-1 function. 
Indeed, the point of this exercise is to locate values for which it 
fails to be 1-1.

The constant 1.064 is just a hair over Zeta[3]^(1/3). I use this because 
it gives a bound on how much larger j2 can be than j1, such that 
DivisorSigma[3,j2]==DivisorSigma[3,j1]. This is not hard to show. It 
also turns out to be reasonably tight: the maximum quotient of such 
j2/j1 seen in this range is around 1.051 (a value oft repeated).

To go further than 10^8 will require a smarter method at least for 32 
bit versions of Mathematica. This run below gets us pretty close to out 
of memory on the machine I used.

In[4]:= InputFormTiming[duplicatedDivisorSumCubes[10^8]]]

Out[5]//InputForm=
{5226.374627, {{194315, 184926, 7401260364550416},
   {295301, 291741, 25751423829890304}, {590602, 583482, 
231762814469012736},
   {1181204, 1166964, 1879853939581992192}, {1476505, 1458705,
    3244679402566178304}, {1886920, 1880574, 7760807890757026560},
   {2067107, 2042187, 8858489797482264576}, {2362408, 2333928,
    15064582940485827840}, {2526095, 2404038, 16267970281281814368},
   {2953010, 2917410, 29202114623095604736}, {3248311, 3209151,
    34300896541413884928}, {3691985, 3513594, 50772646100815853760},
   {3838913, 3792633, 56601629578098888192}, {4134214, 4084374,
    79726408177340381184}, {4469245, 4253298, 90058536115849461888},
   {4724816, 4667856, 120542414947716513024},
   {5020117, 4959597, 126542496700080953856},
   {5610719, 5543079, 176654767473047485440},
   {5635135, 5362854, 180516740291384646240},
   {5906020, 5834820, 236861596387331016192},
   {6023765, 5732706, 220498348780685993472},
   {6496622, 6418302, 308708068872724964352},
   {6791923, 6710043, 313343325162105219072},
   {7382525, 7293525, 405610676744602178304},
   {7677826, 7585266, 509414666202889993728},
   {7966915, 7581966, 510109666845543771552},
   {8268428, 8168748, 646669755216205314048},
   {8355545, 7951818, 588459409064674475328},
   {8563729, 8460489, 628077227211024514560},
   {9132805, 8691522, 768428456089082390784},
   {9449632, 9335712, 964365071005561994496}, {10040234, 9919194,
    1138882470300728584704}, {10298695, 9801078, 1101884840553536833248},
   {10335535, 10210935, 1116169714482765336576},
   {10926137, 10794417, 1304412622679263458816},
   {11221438, 11086158, 1589892907257427368960},
   {11464585, 10910634, 1520070853671364438080},
   {11812040, 11669640, 1898137450501214307840},
   {11853215, 11280486, 1679952880066382524512},
   {12107341, 11961381, 1774839633203699532288},
   {12697943, 12544863, 2047444205866918290432},
   {12993244, 12836604, 2503965447523213599744},
   {13019105, 12390042, 2226032672283641317824},
   {13583846, 13420086, 2820089926458946971648},
   {13796365, 13129746, 2648999899596968491392},
   {14184995, 13499598, 2879223504496673731488},
   {14469749, 14295309, 3038487751960246639872},
   {14765050, 14587050, 3650496090701419604736},
   {15350885, 14609154, 3649117410137937104640},
   {15355652, 15170532, 4131918959201218838016},
   {15650953, 15462273, 3833820476946408678912},
   {16128145, 15348858, 4231951861325553263808},
   {16241555, 16045755, 4321912964218149500928},
   {16536856, 16337496, 5182216531527124776960},
   {17127458, 16920978, 5652695044899220631040},
   {17294035, 16458414, 5217666519197106767520},
   {17422759, 17212719, 5288827426182870635520},
   {18848555, 17937822, 6754937901955686372384},
   {18899264, 18671424, 7714946319468325846272},
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   {19625815, 18677526, 7625533356117022705632},
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   {20671070, 20421870, 10045527430344888029184},
   {20756120, 20686314, 10337396110488359377920},
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   {21556973, 21297093, 10017767395456266281472},
   {21852274, 21588834, 11739713604113371129344},
   {21957595, 20896638, 10679263777489066145568},
   {22442876, 22172316, 12895798025532466437120},
   {22738177, 22464057, 11799508410246376415232},
   {23328779, 23047539, 12696482005089115484160},
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   {24214682, 23922762, 15973556698833295790592},
   {24509983, 24214503, 14724355128845317143552},
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   {29341565, 27923826, 25482184174649583868032},
   {29530100, 29174100, 29609579402355959016192},
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   {89771504, 88689264, 826920966541335279344640},
   {89967845, 85620738, 734596169971147027584768},
   {90657407, 89564487, 745103130726226527230976},
   {90745105, 86360442, 753800338659211824786624},
   {90952708, 89856228, 861364113947985478311936},
   {91838611, 90731451, 774608803133428878870528},
   {92429213, 91314933, 789649334244057589166592},
   {92724514, 91606674, 896897525136745572969984},
   {93076885, 88579554, 813415092887307311331840},
   {93315116, 92190156, 926843186371505430343680},
   {93610417, 92481897, 820311966621089252384256},
   {94201019, 93065379, 836598866645084653393920},
   {94496320, 93357120, 972083236253009056630272},
   {94631405, 90058962, 854855223349088421742464},
   {95086922, 93940602, 970110934701877758246912},
   {95382223, 94232343, 868081527362555343452160},
   {95408665, 90798666, 876092903124834174841152},
   {95972825, 94815825, 891532267484635587912192},
   {96268126, 95107566, 1003706345236047968406528},
   {96858728, 95691048, 1038281185424164226388480},
   {96963185, 92278074, 919617702185936013624000},
   {97740445, 93017778, 941910505639252074027648},
   {97744631, 96566271, 933867453752432268346368},
   {98039932, 96858012, 1074877924405708151479296},
   {98630534, 97441494, 1079426964928105933461504},
   {98906335, 94127334, 976020709083876305347680},
   {98925835, 97733235, 975882755833414051424256},
   {99516437, 98316717, 985577909391129451977216},
   {99811738, 98608458, 1119184253410563785233152}}}


Daniel Lichtblau
Wolfram Research


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