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Homesphenic number

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# sphenic number

Given three primes $p<q<r$, the composite integer $pqr$ is a sphenic number. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, $\ldots$ listed in A007304 of Sloane’s OEIS.

The divisors of a sphenic number therefore are $1,p,q,r,pq,pr,qr,pqr$. Furthermore, $\mu(pqr)=(-1)^{3}$ (where $\mu$ is the Möbius function), $\tau(pqr)=8$ (where $\tau$ is the divisor function) and $\Omega(pqr)=\omega(pqr)=3$ (where $\Omega$ and $\omega$ are the number of (nondistinct) prime factors function and the number of distinct prime factors function, respectively).

The largest known sphenic number at any time is usually the product of the three largest known Mersenne primes.

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