Section 19.3 The Size of the Sum of Divisors Function
For the rest of this chapter, we will focus onxxxxxxxxxx
def _(n=range_slider(1,150,1,(1,20))):
top = n[1]
bottom = n[0]
cols = ((top-bottom)//10)+1
T = [cols*['$n$',r'$\sigma(n)$',r'$\sigma(n)/n$']]
list = [[i,sigma(i),(sigma(i)/i).n(digits=3)] for i in range(bottom,top+1)]
list.extend((10-(len(list)%10))*['',''])
for k in range(10):
t = [item for j in range(cols) for item in list[k+10*j]]
T.append(t)
pretty_print(html(table(T,header_row = True, frame = True)))
Question 19.3.1.
For any given
xxxxxxxxxx
def _(n=[1..15]):
pretty_print(html(r"Try $2^{%s}\cdot3^{%s}\cdot5^{%s}=%s$"%(n, n, n, 2^n*3^n*5^n)))
pretty_print(html(r"Then $\sigma(%s)=%s=%s\cdot %s\approx %s\cdot %s$"%(2^n*3^n*5^n, sigma(2^n*3^n*5^n), sigma(2^n*3^n*5^n)/(2^n*3^n*5^n), 2^n*3^n*5^n, (sigma(2^n*3^n*5^n)/(2^n*3^n*5^n)).n(digits=3), 2^n*3^n*5^n)))
xxxxxxxxxx
print((sigma(6^10)/(6^10)).n())
print((sigma(5*6^10)/(5*6^10)).n())
xxxxxxxxxx
print((sigma(2^4*3^4*5^4*7)/(2^4*3^4*5^4*7)).n(digits=3))
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N = prod([p^4 for p in primes_first_n(100)])
print((sigma(N)/N).n(digits=3))
Fact 19.3.2.
For any positive