Bellman, Richard; Shapiro, Harold N. On a problem in additive number theory. (English) Zbl 0031.25401 Ann. Math. (2) 49, 333-340 (1948). Reviewer: Edmund Hlawka (Wien) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 4 ReviewsCited in 32 Documents MSC: 11N56 Rate of growth of arithmetic functions Keywords:Number theory PDFBibTeX XMLCite \textit{R. Bellman} and \textit{H. N. Shapiro}, Ann. Math. (2) 49, 333--340 (1948; Zbl 0031.25401) Full Text: DOI Online Encyclopedia of Integer Sequences: d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. 1’s-counting sequence: number of 1’s in binary expansion of n (or the binary weight of n). Total number of 1’s in binary expansions of 0, ..., n. a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n. a(n) = tau(tau(n)). a(n) = d(d(d(n))), the 3rd iterate of the number-of-divisors function with an initial value of n. Sum of bits of sum of bits of n: a(n) = wt(wt(n)). a(n) = Sum_{k=1..n} d(d(k)), where d(k) = number of divisors of k. a(n) = sum{k = 1 to n} d(d(d(k))), where d(k) is the number of divisors of k (A000005). Partial sums of A054868.