MathDB

Let

n>1

be an integer and

d_1<d_2<\dots<d_m

the list of its positive divisors, including

1

and

n

. The

m

instruments of a mathematical orchestra will play a musical piece for

m

seconds, where the instrument

i

will play a note of tone

d_i

during

s_i

seconds (not necessarily consecutive), where

d_i

and

s_i

are positive integers. This piece has "sonority"

S=s_1+s_2+\dots s_n

.
A pair of tones

a

and

b

are harmonic if

\frac ab

\frac ba

is an integer. If every instrument plays for at least one second and every pair of notes that sound at the same time are harmonic, show that the maximum sonority achievable is a composite number.

Problem Statement