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))
