2

Part of 1985 Kurschak Competition

Problems(1)

Silly construction problem - power sum of n

Source: Kürschák 1985, problem 2

n\in\mathbb{N}

, define the power sum of

n

as follows. For every prime divisor

p

n

, consider the largest positive integer

k

p^k\le n

, and sum up all the

p^k

's. (For instance, the power sum of

100

2^6+5^2=89

.) Prove that the power sum of

n

n

for infinitely many positive integers

n

number theory unsolvednumber theory