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Colossally abundant number with prime factors 2 to 999,781,070,909

The program superabun3842a.out finished around January 6th. It searched for highly abundant numbers. The output was re-directed to a file. Afterwards, I wanted to check independently the values of log(N) and sigma(N)/N found by the program. Each N is given in shortened prime factorization form. In this form, adequate with non-increasing exponents for the primes, only the starting index and/or ending index for some exponent are given.

For example,

1 44
2 27
3 18
4 15
5 12
6 11
7 10
9  9
10 8
13 7
20 6
33 5
69 4
222 3
1673 2
106913 1

37599987402 0

is the number N where 2 has exponent 44, 3 has exponent 27, 5 has exponent 18, 7 has exponent 15, 11 has exponent 12, 13 has exponent 11, 17 and 19 have exponent 10. The number 19 is the 8th prime, and its exponent in the factorization of N is 10, the same as for the 7th prime 17. So we skip 8, and at the 9th prime, 23, the exponent is 9. For the 10th prime, the exponent is 8. The 11th and 12th primes also have exponent 8 in N. The 13th through the 19th primes have exponent 7 in N. The 20th through the 32nd primes have exponent 6 in N. The 33rd through the 68th primes have exponent 5 in N. The 69th through the 221st primes have exponent 4 in N. The 222nd through the 1672nd prime have exponent 3 in N. The 1673rd through the 106912nd primes have exponent 2 in N. The 106913rd through the 37599987401st primes have exponent 1 in N. The primes starting at the 37599987402 do not divide N. According to my calculations, the 37599987401st prime is 999,781,070,909 or just under 10^12 .  It took 40 hours independently to sum the logarithms of the primes from

2 to 10^12:

[david2@localhost eratosthenes7]$ time ./eratosthsiv3630a.out

size of int is: 4
size of long is: 8
size of char is: 1

pi(1000000000000) = 37607912018

sum_logs_primes = 999999030333.09622463699607903975308725646673
real 2365m37.094s
user 2364m50.247s
sys 0m2.488s

[david2@localhost eratosthenes7]$

The following command in the PARI/gp calculator,

? count = 37599987401; forprime(q=999781070910 , 1000000000000, count=count+1); count

%196 = 37607912018

implies that if pi(10^12) = 37607912018 then pi(999781070910) = 37599987401. Since 999781070909 is the last prime below 999781070910, 999781070909 is the 37599987401st prime.

Below, the PARI/gp command to sum the logs of primes from 999781070910 to 10^12:

? sumlogprimes=0; forprime(q=999781070910 , 1000000000000, sumlogprimes=sumlogprimes+log(q)); sumlogprimes
%199 = 218964392.249194594870107709322349966327713173580

Then, after Tchebychev,

? thetaCheb = 999999030333.09622463699607903975308725646673-sumlogprimes
%201 = 999780065940.847030042125971330430737290139016826

That is the sum of the logs of primes from 2 to 999781070909 inclusive, calculated in PARI/gp.

Then we calculate the critical exponent (CA numbers) for 999781070909: the largest epsilon for which 999781070909 has exponent 1 in C.A. numbers.

? p=999781070909
%204 = 999781070909
? epsilon
%205 = 8.6159474458768635875553049472001443044 E-15
? epsilon=log(1+1/p)/log(p)
%206 = 3.61994187188332908418733101506308857973891597465 E-14  <=== critical exponent where

999781070909 first attains exponent 1.

g =
(Y)->floor(log((exp((1+epsilon)*log(Y))-1)/(exp(epsilon*log(Y))-1))/log(Y))-1

as a PARI/gp function implements the Erdos-Alaoglu, or Erdos-Alaoglu-Ramanujan formula for the exponent of the prime Y for exponent or critical exponent epsilon.

? g(2)
%207 = 44
? g(17)
%208 = 10
? g(19)
%209 = 10

Next, we calculate contribution to log(N) of exponents larger than 1,

for primes with index 1 [that is , the prime two] to index 106912 inclusive:

? sum4=0; for(X=1,106912, sum4 = sum4+(g(prime(X))-1)*log(prime(X)))
? sum4
%215 = 1412045.01159541398434404224867771016684988078484

? prime(106912)
%212 = 1397233
? g(1397233)
%213 = 2    OK

? prime(106913)
%210 = 1397251
? g(1397251)
%211 = 1  Ok.

Then we need to add thetaCheb and sum4  to get log(N):

? thetaCheb+sum4
%217 = 999781477985.858625456110315372679415000305866707

The abundant numbers program is very long, and was developped in stages. It prints values of log(log(N)) for highly abundant N.

? log(thetaCheb+sum4)
%218 = 27.6308025700349926529916290589321648678391771498

The above is log(log(N)) for the colossally abundant N with critical exponent:

? epsilon=log(1+1/p)/log(p)
%206 = 3.61994187188332908418733101506308857973891597465 E-14

where

p=999781070909 , according to the check-program (short, simple).

The value of log(log(N)) computed by superabun3842a.out is marked by (*****) :

current step number = 17
quotient de Robin: 0.999999972115680424387739328044
loglog n = 27.630802570034992652991629058932                        (*****)
sigma(n’)/n’ = 49.212458972166029828348808887298
record step is at: 37599987401
log(log(n’)) = 27.630802570034992652991629058932                    (*****)
PARI/GP
?
log(thetaCheb+sum4)
%218 = 27.6308025700349926529916290589321648678391771498

log(log(n’)) = 27.630802570034992652991629058932 30 digits OK .

Robin upper bound = 49.212460344422001174357312333199
sigma(n’)/n’ = 49.212458972166029828348808887298

I still  would like to verify independently sigma(N)/N.

=============================================

==============================================

=================================
step at 1
step at 2
step at 3
step at 4
step at 5
step at 6
step at 7
step at 9
step at 10
step at 13
step at 20
step at 33
step at 69
step at 222
step at 1673
step at 106913
step at 37599987401

? for(X=1,1, print(X,” “, g(prime(X))));for(X=2,1000000, if(g(prime(X))<g(prime(X-1)),print(X,” “,g(prime(X)))))

1 44
2 27
3 18
4 15
5 12
6 11
7 10
9 9
10 8
13 7
20 6
33 5
69 4
222 3
1673 2
106913 1

time = 4min, 41,736 ms.
—————————————

? p2=999781070909
%194 = 999781070909

? factor(999781070909)
%195 =
[999781070909 1]
? count = 37599987401; forprime(q=999781070910 , 1000000000000, count=count+1); count
time = 2,294 ms.
%196 = 37607912018

? 37607912018
%197 = 37607912018

? p2
%198 = 999781070909
? factor(999781070909)
%195 =
[999781070909 1]

? count = 37599987401; forprime(q=999781070910 , 1000000000000, count=count+1); count
time = 2,294 ms.
%196 = 37607912018
? 37607912018
%197 = 37607912018
? p2
%198 = 999781070909
? sumlogprimes=0; forprime(q=999781070910 , 1000000000000, sumlogprimes=sumlogprimes+log(q)); sumlogprimes
time = 1min, 25,717 ms.
%199 = 218964392.249194594870107709322349966327713173580
? 999999030333.09622463699607903975308725646673 – sumlogprimes
%200 = 999780065940.847030042125971330430737290139016826
? thetaCheb = 999999030333.09622463699607903975308725646673-sumlogprimes
%201 = 999780065940.847030042125971330430737290139016826
? thetaCheb
%202 = 999780065940.847030042125971330430737290139016826
? p
%203 = 3999819583387
? p=999781070909
%204 = 999781070909
? epsilon
%205 = 8.6159474458768635875553049472001443044 E-15
? epsilon=log(1+1/p)/log(p)
%206 = 3.61994187188332908418733101506308857973891597465 E-14
? \u
g =
(Y)->floor(log((exp((1+epsilon)*log(Y))-1)/(exp(epsilon*log(Y))-1))/log(Y))-1

? g(2)
%207 = 44
? g(17)
%208 = 10
? g(19)
%209 = 10
? prime(106913)
%210 = 1397251
? g(1397251)
%211 = 1
? prime(106912)
%212 = 1397233
? g(1397233)
%213 = 2
? sum=0; for(X=1,106912, sum = sum+(g(prime(X)-1)*log(prime(X)));sum
*** syntax error, unexpected $end, expecting )-> or ‘,’ or ‘)’: …e(X)-1)*log(prime(X)));sum
*** ^-
? sum=0; for(X=1,106912, sum = sum+(g(prime(X))-1)*log(prime(X)))
*** variable name expected: sum=0;for(X=1,106912
*** ^——————–
? sum4=0; for(X=1,106912, sum4 = sum4+(g(prime(X))-1)*log(prime(X)))
time = 7,518 ms.
? sum4
%215 = 1412045.01159541398434404224867771016684988078484
? thetaCheb
%216 = 999780065940.847030042125971330430737290139016826
? thetaCheb+sum4
%217 = 999781477985.858625456110315372679415000305866707
? log(thetaCheb+sum4)
%218 = 27.6308025700349926529916290589321648678391771498
? 37599987401/80000000.0000
%219 = 469.999842512500000000000000000000000000000000000
? 469+1
%220 = 470
? 470*80000000
%221 = 37600000000

[david2@localhost eratosthenes7]$ time ./eratosthsiv3630a.out

size of int is: 4
size of long is: 8
size of char is: 1

pi(1000000000000) = 37607912018

sum_logs_primes = 999999030333.09622463699607903975308725646673
real 2365m37.094s
user 2364m50.247s
sys 0m2.488s

[david2@localhost eratosthenes7]$
—————————————

Number of big primes used in making current n is: 37598987400
numloops = 799999

number of steps: 17
44 27 18 15 12 11 10 10 9 8 8 8 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
record_low_variation_delta = -0.006989192976654998649573742516 record_low_log2n = 27.630802567271308507202087382619

current step number = 1 quotient de Robin: 0.999999972115669543306341175815
loglog n = 27.630802570008049110251729402793

sigma(n’)/n’ = 49.212458972117505944083146994669
record step is at: 1
log(log(n’)) = 27.630802570008049110251729402793 Robin upper bound = 49.212460344374012773540337008566 sigma(n’)/n’ = 49.212458972117505944083146994669
current step number = 2 quotient de Robin: 0.999999972115669796471084435642
loglog n = 27.630802570008454663982266449126

sigma(n’)/n’ = 49.212458972118240723486366399751
record step is at: 2
log(log(n’)) = 27.630802570008454663982266449126 Robin upper bound = 49.212460344374735094103809570873 sigma(n’)/n’ = 49.212458972118240723486366399751
current step number = 3 quotient de Robin: 0.999999972115664106313899496870
loglog n = 27.630802570008965601257088259561

sigma(n’)/n’ = 49.212458972118870713113682345841
record step is at: 2
log(log(n’)) = 27.630802570008965601257088259561 Robin upper bound = 49.212460344375645110391317775089 sigma(n’)/n’ = 49.212458972118870713113682345841
current step number = 4 quotient de Robin: 0.999999972115635775153723508512
loglog n = 27.630802570009302147036379714800

sigma(n’)/n’ = 49.212458972118075879405184158090
record step is at: 2
log(log(n’)) = 27.630802570009302147036379714800 Robin upper bound = 49.212460344376244522796204802772 sigma(n’)/n’ = 49.212458972118075879405184158090
current step number = 5 quotient de Robin: 0.999999972115619954714472491358
loglog n = 27.630802570009754230950421738977

sigma(n’)/n’ = 49.212458972118102510833377823037
record step is at: 2
log(log(n’)) = 27.630802570009754230950421738977 Robin upper bound = 49.212460344377049716986122103593 sigma(n’)/n’ = 49.212458972118102510833377823037
current step number = 6 quotient de Robin: 0.999999972115627194622094540962
loglog n = 27.630802570009921321548063048059

sigma(n’)/n’ = 49.212458972118756404946590923917
record step is at: 2
log(log(n’)) = 27.630802570009921321548063048059 Robin upper bound = 49.212460344377347317440886537232 sigma(n’)/n’ = 49.212458972118756404946590923917
current step number = 7 quotient de Robin: 0.999999972115605325383227100624
loglog n = 27.630802570010189644169063997968

sigma(n’)/n’ = 49.212458972118158067902126921320
record step is at: 2
log(log(n’)) = 27.630802570010189644169063997968 Robin upper bound = 49.212460344377825219460274166564 sigma(n’)/n’ = 49.212458972118158067902126921320
current step number = 8 quotient de Robin: 0.999999972115590010552154801447
loglog n = 27.630802570010491991110406975108

sigma(n’)/n’ = 49.212458972117942889168174424680
record step is at: 2
log(log(n’)) = 27.630802570010491991110406975108 Robin upper bound = 49.212460344378363721258163843418 sigma(n’)/n’ = 49.212458972117942889168174424680
current step number = 9 quotient de Robin: 0.999999972115625084463154265373
loglog n = 27.630802570010723843389296918164

sigma(n’)/n’ = 49.212458972120081908309822115148
record step is at: 2
log(log(n’)) = 27.630802570010723843389296918164 Robin upper bound = 49.212460344378776666957142892032 sigma(n’)/n’ = 49.212458972120081908309822115148
current step number = 10 quotient de Robin: 0.999999972115668176324314499768
loglog n = 27.630802570011070195311542746581

sigma(n’)/n’ = 49.212458972122819442656764330207
record step is at: 2
log(log(n’)) = 27.630802570011070195311542746581 Robin upper bound = 49.212460344379393544812772822695 sigma(n’)/n’ = 49.212458972122819442656764330207
current step number = 11 quotient de Robin: 0.999999972115634517989633902184
loglog n = 27.630802570011619423140264094097

sigma(n’)/n’ = 49.212458972122141247705688916999
record step is at: 2
log(log(n’)) = 27.630802570011619423140264094097 Robin upper bound = 49.212460344380371759349701059424 sigma(n’)/n’ = 49.212458972122141247705688916999
current step number = 12 quotient de Robin: 0.999999972115652463082734815002
loglog n = 27.630802570012276867855209347208

sigma(n’)/n’ = 49.212458972124195326503784859111
record step is at: 2
log(log(n’)) = 27.630802570012276867855209347208 Robin upper bound = 49.212460344381542715997843477838 sigma(n’)/n’ = 49.212458972124195326503784859111
current step number = 13 quotient de Robin: 0.999999972115666879890460152981
loglog n = 27.630802570013206414835827203974

sigma(n’)/n’ = 49.212458972126560403524500125030
record step is at: 2
log(log(n’)) = 27.630802570013206414835827203974 Robin upper bound = 49.212460344383198306486248011687 sigma(n’)/n’ = 49.212458972126560403524500125030
current step number = 14 quotient de Robin: 0.999999972115679079789989644612
loglog n = 27.630802570014600907757977282106

sigma(n’)/n’ = 49.212458972129644483407768658400
record step is at: 222
log(log(n’)) = 27.630802570014600907757977282106 Robin upper bound = 49.212460344385681999366972068265 sigma(n’)/n’ = 49.212458972129644483407768658400
current step number = 15 quotient de Robin: 0.999999972115680283018417941615
loglog n = 27.630802570016921922813175727773

sigma(n’)/n’ = 49.212458972133837593020364535794
record step is at: 1673
log(log(n’)) = 27.630802570016921922813175727773 Robin upper bound = 49.212460344389815895263526016736 sigma(n’)/n’ = 49.212458972133837593020364535794
current step number = 16 quotient de Robin: 0.999999972115680415057292588568
loglog n = 27.630802570021508921629356897342

sigma(n’)/n’ = 49.212458972142013867823292313363
record step is at: 106913
log(log(n’)) = 27.630802570021508921629356897342 Robin upper bound = 49.212460344397985672336379988117 sigma(n’)/n’ = 49.212458972142013867823292313363
current step number = 17
quotient de Robin: 0.999999972115680424387739328044
loglog n = 27.630802570034992652991629058932
sigma(n’)/n’ = 49.212458972166029828348808887298
record step is at: 37599987401
log(log(n’)) = 27.630802570034992652991629058932
PARI/GP
?
log(thetaCheb+sum4)
%218 = 27.6308025700349926529916290589321648678391771498

log(log(n’)) = 27.630802570034992652991629058932 30 digits OK .

Robin upper bound = 49.212460344422001174357312333199
sigma(n’)/n’ = 49.212458972166029828348808887298
=================================

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Written by meditationatae

January 15, 2015 at 8:19 am

Posted in History

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