Vol. 57, No. 2, 1975

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Some new results on odd perfect numbers

G. G. Dandapat, John L. Hunsucker and Carl Pomerance

Vol. 57 (1975), No. 2, 359–364

If m is a multiply perfect number (σ(m) = tm for some integer t), we ask if there is a prime p with m = pan, (pa,n) = 1(n) = pa, and σ(pa) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers necessarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) = 2n.

Mathematical Subject Classification
Primary: 10A40
Received: 16 October 1974
Published: 1 April 1975
G. G. Dandapat
John L. Hunsucker
Carl Pomerance
Mathematics Department
Dartmouth College
Kemeny Hall
Dartmouth College
Hanover NH 03755
United States