THE SUM OF DIGITS OF n AND n²

Abstract

Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 2005, Melfi examined the structure of n such that s 2 (n) = s 2 (n 2 ). We extend this study to the more general case of generic q and polynomials p(n), and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case p(n) = n 2 , looking at the subsets of n where s q (n) = s q (n 2 ) = k for fixed k.

Sums of digits in q-ary expansions

“…In other words, can we say anything about #{n ≤ N | sq(n 2 ) sq (n) = a c } for some positive rational a c ? This has been done for the special case a c = 1 in [3] and [5]. This also leads to the major open question posed by Stolarsky of obtaining the average value of this ratio.…”

“…Notice n ≥ k + 2 because 3c < 6a and so 3c + 3 ≤ 6a so that 3(c − a) + 2 ≤ 3a − 1. By Lemma 3.10, we have for (u) 3 …”

“…This lower bound was improved to N 1/19 by Hare, Laishram, and Stoll [3]. Other work was done by Drmota and Rivat [1] in studying the binary digits of squares.…”

Squares with Three Nonzero Digits

“…On the other hand, in the case q = 2 (where S q (n) and N q (n) coincide), Stolarsky [20] proved that, for infinitely many n, N 2 (n 2 ) N 2 (n) ≤ 4 (log log n) 2 log n , a result that was subsequently substantially sharpened and generalized by Hare, Laishram and Stoll [13]. Further developments are well described in [14] where, in Date: October 12, 2018. 1991 Mathematics Subject Classification.…”

Products of integers with few nonzero digits

“…However, it is far from clear for which triples (k, ℓ, m) such solutions exist, or even more generally, whether there are finitely of infinitely many solutions in odd a, b for the system (1) for a given triple (k, ℓ, m). Part of the motivation to study (1) also comes from a paper of Hare, Laishram, Stoll [8] where the authors studied the solution set of the equation s(a 2 ) = s(a) = k, which is a particular instance of (1). For example, they showed that s(a 2 ) = s(a) = 8 only allows finitely many odd solutions a, whereas s(a 2 ) = s(a) = 12 has infinitely many odd solutions a.…”