The sum of a digitaddition series
Kenneth B. Stolarsky · Proceedings of the American Mathematical Society · 1976
Let B ( x ) B(x) be the number of ones in the binary expansion of x x . A “digitaddition series” is a sequence y 1 > y 2 > y 3 > … {y_1} > {y_2} > {y_3} > \ldots , where y 1 {y_1} is a given positive integer and y n + 1 = y n + B ( y n ) for n = 1 , 2 , … {y_{n + 1}} = {y_n} + B({y_n})\;{\text {for}}\;n = 1,2, \ldots . Various questions involving the y m {y_m} are studied; in particular, the asymptotic result y m ∼ ( m log m ) / ( 2 log 2 ) {y_m} \sim (m\log m)/(2\log 2) is proved.