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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, repetition
included 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) = 210.

The number ‘new’ has an unusually high
sum of the divisors function value, also
denoted sigma_1(.), and this is exemplified
by ‘new’ (denoted ‘n’ in the formula)
having the property:

       sigma_1(n)
—————————
 H(n) + log(H(n))*exp(H(n))

~=  0.9993847971371  .

The computations were done with PARI/gp .

? P(W) = prod(X=1,W,prime(X));

? new = (2^10)*(3^3)*P(3)*P(3)*P(4)*P(6)*P(12)*P(42)*P(2170) ;

? sigma(new,1)/(harmonic(new)+log(harmonic(new))*exp(harmonic(new)))
%594 = 0.99938479713714118220281058652853988770

? floor(log(new)/log(10))+1
%597 = 8307

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

? \u
P =
  (W)->prod(X=1,W,prime(X))

Qr =
  (W)->sigma(W,1)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))

harmonic =
  (Z)->Euler+psi(Z+1)

primorial =
  (W)->prod(X=1,W,prime(X))

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Written by meditationatae

June 18, 2013 at 6:02 am

Posted in History

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