Triples of Consecutive Sphenic Numbers

Theorem

The sequence of triplets of consecutive sphenic numbers starts:

$\tuple {1309, 1310, 1311}, \tuple {1885, 1886, 1887}, \tuple {2013, 2014, 2015}, \ldots$

Note that there cannot be quadruplets of such numbers, since one of the quadruplets must be divisible by $4$, making it non-sphenic.

We have:

$\ds 1309$

$=$

$\ds 7 \times 11 \times 17$

$\ds 1310$

$=$

$\ds 2 \times 5 \times 131$

$\ds 1311$

$=$

$\ds 3 \times 19 \times 23$

$\ds 1885$

$=$

$\ds 5 \times 13 \times 29$

$\ds 1886$

$=$

$\ds 2 \times 23 \times 41$

$\ds 1887$

$=$

$\ds 3 \times 17 \times 37$

$\ds 2013$

$=$

$\ds 3 \times 11 \times 61$

$\ds 2014$

$=$

$\ds 2 \times 19 \times 53$

$\ds 2015$

$=$

$\ds 5 \times 13 \times 31$

hence each number above is sphenic.

$\blacksquare$