## 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))