On the equation σ*(σ*(n)) = 2n
Abstract
As usual σ(n) denotes the sum of the divisors of n and σ*(n) the sum of the unitary divisors of n. Suryanarayana defined super perfect numbers as solutions of σ(σ(n)) = 2n and showed that even super perfect numbers are of the form n = 2k provided 2k+1-1 is a prime. He asked for the existence of odd super perfect numbers. J.L. Hunsucker and Carl Pomerance (Indian J. Math. 17 (1975)) showed that there are no such numbers less than 7 · 1024. In this paper we consider solutions of σ*(σ*(n)) = 2n which may be called unitary super perfect numbers (USP numbers). While σ*(σ*(n)) = 2n + 1 has no solutions and σ*(σ*(n)) = 2n - 1 has n = 1, 3 as the only solutions, there are both even and odd USP numbers - ten of them up to 24000 (namely 2, 9, 165, 238, 1640, 4320, 10250, 100824, 13500 and 23760) and 22 of them upto 108 as listed in the appendix at the end. We do not know of any odd USP numbers other than 9 and 165. Perhaps such numbers are finite in number whereas even USP numbers are likely to be infinite.
