Descartes number

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In mathematics, a Descartes number is a number which is close to being a perfect number. They are named for René Descartes who observed that the number D = 32⋅72⋅112⋅132⋅22021 = 198585576189 would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for D satisfies

\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^2+13+1)\cdot(22021+1) \ .

A Descartes number is defined as an odd number n = mp where m and p are coprime and 2n = σ(m)⋅(p+1). The example given is the only one currently known.

If m is an odd almost perfect number, that is, σ(m) = 2m−1, then m(2m−1) is a Descartes number.

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