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Superabundant numbers and number-theoretical computations with PARI/gp

This is an update, showing my latest PARI/gp “optimized” command-line to search for integers with “extreme” abundancy; one in a series of recent posts on Gronwall’s Theorem, the Theorem of Guy Robin, the Lagarias criterion for RH and the works of Ramanujan and Erdos&Alaoglu.

New sample PARI/gp command line using 2 “iterations” of “outer loop” :
__________________________________________________________________________

? for(YYY = 1, 2  ,   for(ZZZZ=1,5,   thebest = 0; www3=vector(100);for(Y=1,100,www3[Y]=www[Y]);stdwww3=vector(60000);for(Y=1,60000, stdwww3[Y] = sum(Z=1,22, www3[Z] > (Y-1)));for(Y= 1001,49000, stdwww3[Y] = 0);        cst=ratiodd(stdwww3);  print(”          “,cst);   for(WW = 1,     1     ,   for(X=1,22,  www2=vector(100);for(Y=1,100,www2[Y]=www[Y]);www2[X]=www2[X]+1;  stdwww2=vector(60000);for(Y=1,60000, stdwww2[Y] = sum(Z=1,22,www2[Z] > (Y-1)));  for(Y=1001,49000,stdwww2[Y]=0); newr2 = ratiodd(stdwww2);  if(newr2>   cst         ,  print(X,”         “, newr2));   if(newr2>cst, cst=newr2; thebest = X)                  ) ) ; print(“Best:  “,thebest,”  Score:  “,cst); if(thebest>0, www[thebest]=www[thebest]+1)   ) ; print(www) )   

Ouput for preceding command-line in full (2 iterations only of “outer loop”):
_____________________________________________________________________________

          9999.2344741160447345088252272119753296
1         9999.2344888925751966186187116790815267
Best:  1  Score:  9999.2344888925751966186187116790815267
          9999.2344888925751966186187116790815267
1         9999.2345038764507951834068751314938346
Best:  1  Score:  9999.2345038764507951834068751314938346
          9999.2345038764507951834068751314938346
1         9999.2345188974171339310642820346303811
Best:  1  Score:  9999.2345188974171339310642820346303811
          9999.2345188974171339310642820346303811
1         9999.2345342391980490149115736647308356
Best:  1  Score:  9999.2345342391980490149115736647308356
          9999.2345342391980490149115736647308356
1         9999.2345498450316425004958058900632011
Best:  1  Score:  9999.2345498450316425004958058900632011
[50269, 180, 30, 12, 8, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
          9999.2345498450316425004958058900632011
1         9999.2345654879275262799988338463734536
Best:  1  Score:  9999.2345654879275262799988338463734536
          9999.2345654879275262799988338463734536
1         9999.2345811678831764734681248703828065
Best:  1  Score:  9999.2345811678831764734681248703828065
          9999.2345811678831764734681248703828065
1         9999.2345967146816410017570100839477279
Best:  1  Score:  9999.2345967146816410017570100839477279
          9999.2345967146816410017570100839477279
1         9999.2346121283372364773890360174238560
Best:  1  Score:  9999.2346121283372364773890360174238560
          9999.2346121283372364773890360174238560
1         9999.2346273521334352581443880153001546
Best:  1  Score:  9999.2346273521334352581443880153001546
[50274, 180, 30, 12, 8, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

{ Nota Bene: ouput complete.}

The 100-element vector named `www’ :
__________________________________________________________________________

? www
%216 = [50274, 180, 30, 12, 8, 5, 4, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

The user-defined functions `harmonic’ and `ratiodd’ :
__________________________________________________________________________

? \u
harmonic =
  (Z)->Euler+psi(Z+1)

…….skip_some_output…….

ratiodd =
  (ZZZ)->base*(sigcpp48k*prod(Y=1,1000,sum(W=0,ZZZ[Y],prime(Y)^W))*prod(Y=49001,60000,sum(W=0,ZZZ[Y],prime(Y)^W))/(harmonic(cpp48k*prod(Y=1,1000,prime(Y)^ZZZ[Y])*prod(Y=49001,60000,prime(Y)^ZZZ[Y]))+log(harmonic(cpp48k*prod(Y=1,1000,prime(Y)^ZZZ[Y])*prod(Y=49001,60000,prime(Y)^ZZZ[Y])))*exp(harmonic(cpp48k*prod(Y=1,1000,prime(Y)^ZZZ[Y])*prod(Y=49001,60000,prime(Y)^ZZZ[Y])))))

{ Nota Bene: output of PARI/gp command-line `\u [Enter]’  complete.}

 
Definitions of the integer constants `cpp48k’  and `sigcpp48k :
__________________________________________________________________________

cpp48k = prod(X= 1001 ,49000, prime(X));

sigcpp48k = prod(X=1001,49000, 1+prime(X));

Value of the constant `base’ :
__________________________________________________________________________

? base
%217 = 10000

Comments pertaining to super-colossal numbers (Erdod/Alaoglu, ca. 1946)
__________________________________________________________________________

They were first studied by Ramanujan; unfortunately for his 1915 paper
on highly composite numbers, the journal where he submitted his work
was in financial difficulties, and so Ramanujan agreed to remove aspects
of the work to reduce the cost of printing.

Source: Wikipedia article entitled “Colossally abundant number”.

Also relevant seems to be a 1944 paper by Erdos and Alaoglu.

Source:  The On-Line Encyclopedia of Integer Sequences, pioneered by Neil J. Sloane.

Web Source at OEIS for CA numbers is:  http://oeis.org/A004490 .

Nate Christensen gave a Senior Seminar presentation and/or expository
paper entitled “The Harmonic Series, Colossally Abundant Numbers and the Riemann Hypothesis”
in Spring 2011 at University {Blank,Unknown,?}.

In formula (2.3), he gives a fundamental formula due to Erdos and Alaoglu
that is the key to determining so-called “critical exponents”, i.e.
values of a small parameter epsilon>0  that characterize (as I understand it)
either 1, 2 or 4 CA numbers. Typically, a “critical exponent” might
belong to 1, or sometimes 2 CA numbers. The case of 4 distinct CA numbers
for one “critical exponent” was/is an unsolved problem of Diophantine
Analysis (maybe, but this is not my main topic here, so I might
be wrong).

The Erdos-Alaoglu formula in (2.3) of Nate Christensen’s paper or senior
thesis (etc.) has as parameter a prime p and a small positive
exponent epsilon, which are the determinants of an integer
exponent denoted here a_p (epsilon):

a_p (epsilon) :=
floor((log(p^(1+epsilon)-1) – log(p^epsilon -1 ))/log(p)) – 1

? p
%509 = 2

? epsilon
%510 = 8.599133311962863885918219014490855495185170921\
11310266255836514942466356220054151046798601121983628\
78407220224806951385036093587798592412113339118630107\
72492041978611853131929410681643652755325742540775832\
70712506537440691075699583030496270273937335496515351\
47063898918897076310654895220769089733458518813156101\
13008259123414326535135778237837120928980688873355405\
89989559765060909832106232056751017208975574819838490\
01651681573189315956317649091340695190772197483907245\
442563587930505065060543381152 E-8

? (log(p^(1+epsilon)-1) – log(p^epsilon -1 ))/log(p) – 1
%511 = 23.00000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000
0000000000000 etc.

so then:

floor((log(p^(1+epsilon)-1) – log(p^epsilon -1 ))/log(p)) – 1
will give 23 for p=2.

The value
epsilon =
8.59913331196286388591821901449085549518517 E -8

is then near-critical for passing from a_2 = 22 to a_2 = 23.

? p=851639
%559 = 851639

? (log(p^(1+epsilon)-1) – log(p^epsilon -1 ))/log(p) – 1
%560 = 1.0000000716916720788566294408105580472712241078687514062340430

In terms of products of 22 primorials, currently the prime 2 has
exponent 22.  I suspect that perhaps the algorithm/procedure I use
might halt before reaching numbers with a primorial factor
of about primorial(68000), because:

? prime(67740) =  851639.

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Written by meditationatae

July 12, 2013 at 4:53 pm

Posted in History

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