The latest PARI/gp command-line and first few lines of ouput are copied below. Note that Keith Briggs has already published in Experimental Mathematics around 2006 on superabundant and colossally abundant numbers out to ~~~ 10^(10^10) or so, using a sieve technique to locate the first ~ 10^9 or 10^10 primes, and making use of the… Continue reading My insane pursuit of highly abundant numbers …
Tag: primorials
About a “highly composite” number called “a8″
a8 = primorial(3302)*primorial(52)*primorial(14) *primorial(7)*(primorial(4)^2)*primorial(3) *(primorial(2)^4)*(primorial(1)^7); sigma(a8)/(H(a8)+log(H(a8))*exp(H(a8))) 0.999531374987652061429003800216023447038030443823248117806354
About a 12025-digit “highly composite” number called “a4”
I’ve been looking for whole numbers ‘n’ for whichthe sum of divisors function sigma(n) has a valuethat is “extremely large” relative to n. It can be shown that: limsup n→∞ σ(n)/[n log(log(n)) ] = e^gamma . (Grönwall, 1913) where sigma(n) is the sum of all divisors of the positive integer n. I got a 12,025-digit… Continue reading About a 12025-digit “highly composite” number called “a4”
About an 8307-digit “highly composite” number
The number ‘new’ below has 8307 digits .It is the product of 7 primorial numbers greater than or equal to 30, repetitionincluded in the count, and the two “high-powers” 2^10 and 3^3 ; So, new := (2^10)*(3^3)*P(3)*P(3)*P(4)*P(6)*P(12)*P(42)*P(2170) , where P(k) is the product of the ‘k’ first primes.For example, P(3) = 30 and P(4) =… Continue reading About an 8307-digit “highly composite” number
PARI/gp code related to RH criterion, Lagarias: sigma_1(n) asymptotics
The essential PARI/gp code to check my computation is below: ? primorial(1)%462 = 2? primorial(1)^6%463 = 64? m16%464 = m16? \uharmonic = (Z)->Euler+psi(Z+1) primorial = (W)->prod(X=1,W,prime(X)) ? Qr(W) = sigma(W,1)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))%465 = (W)->sigma(W,1)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))? m16%466 = m16? m16 = primorial(1614)*primorial(37)*primorial(11)*primorial(6)*primorial(4)*primorial(3)*primorial(2)^4*primorial(1)^6 ;? Qr(m16)%468 = 0.99923459761253613955435440267376833156?
Factoring highly composite number as primo. product, Result
? m16 = primorial(1614)*primorial(37)*primorial(11)* primorial(6)*primorial(4)*primorial(3)* primorial(2)^4*primorial(1)^6 ; ? big – m16%245 = 0? ==== done! Ok.
Factoring the Big number earlier as primorials product, Step 2
? (big/primorial(1614))/primorial(37)%233 = 3147213746118215854080000? factor((big/primorial(1614))/primorial(37))%234 = [2 14] [3 8] [5 4] [7 3] [11 2] [13 2] [17 1] [19 1] [23 1] [29 1] [31 1] ?