ad Theorem 1: g(n) is *exactly* the number of integral powers of two that sum to n: Any integer n has a unique representation in any (positive integer) base b>=2 as n=sum_i a_i b^i (i>=0, 0<= a_i <= b-1) So we have n=sum_i a_i 2^i (i>=0, 0<=a_i<=1), and g(n)=sum_i a_i both with unique a_i. Christoph ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of Erich Friedman [erichfriedman68@gmail.com] Sent: Thursday, June 02, 2016 2:11 AM To: math-fun Subject: [math-fun] sum of binary digits Let n be a positive integer. Let g(n) be the sum of the binary digits of n. I am looking for interesting facts involving g(n). Here are 3 that i have so far, in increasing difficulty to prove: Theorem 1: The minimum number of integral powers of 2 to sum to n is g(n). Theorem 2: The highest power of 2 to divide n! is 2^[ n - g(n) ]. Theorem 3: The number of odd entries in the nth row of Pascal's Triangle is 2^[ g(n) ]. Anyone know any others? Erich Friedman _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun