Problems & Puzzles: Problems
Problem 21.- Divisors (III). Certain questions about s(N), the sum of divisors of a number N.
Let's define s(N) as the sum of the positive divisors of N.
Example: s(14) = 1+2+7+14 = 24
a) Can you calculate s(2097727632)? Can you explain your algorithm?
b) What non-trivial can you say about the existence of solutions for s(N) = K, for any positive number K?
c) Can you calculate or - at least - bound the minimal N for a given K, supposed sigma(N) = K has solutions.
d) There are many couples of consecutive numbers N and N+1 such that s(N) = s(N+1). Some of these are 14 & 15, 206 & 207, 957 & 958, etc. (See B13, p. 68 of Richard K. Guy's book "Unsolved Problems in Number Theory". Also the Sloane's sequence A002961). But as far as I know, it hasn't been discovered any triplet of consecutive numbers such that s(N) = s(N+1) = s(N+2). Can you find the first of such triplets? (Hint: Jud McCranie - who has done exhaustive search in this subject also, informs at 22/07/99: "I found no solutions for s(n)=s(n+1)=s(n+2) for n < 4,250,000,000").
a) T.W.A Baumann & Enoch
Haga independently sent the following solution
Enoch sent the following reference for this: Albert Beiler, "Recreations in Theory of Numbers", p. 20.
Dominique Toublanc wrote (April, 2006):