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S(k)=a(1) + a(2) + ... + a(2^k) when n = 2^{a(n)} (2b(n)+1),

Source: Netherlands - Dutch NMO 1971 p4

January 28, 2023

number theorySum

Problem Statement

For every positive integer

n

there exist unambiguously determined non-negative integers

a(n)

and

b(n)

such that

n = 2^{a(n)}(2b(n)+1),

For positive integer

k

we define

S(k)

by:

a(1) + a(2) + ... + a(2^k) = S(k)

Express

S(k)

in terms of

k