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sphenic number (Definition)

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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Cross-references: Mersenne primes, product, number of distinct prime factors function, divisor function, Möbius function, divisors, OEIS, integer, composite, primes
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This is version 4 of sphenic number, born on 2006-08-17, modified 2006-11-15.
Object id is 8262, canonical name is SphenicNumber.
Accessed 1106 times total.

Classification:
AMS MSC11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

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