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Prime power puzzle (Posted on 2017-04-01) Difficulty: 3 of 5
603, 604, and 605 are the first 3 consecutive integers that are the product of a prime and another prime squared.

603=32*67
604=22*151
605=5*112

1. What is the first set of 4 consecutive integers that are the product of a prime and another prime squared?
2. What is the first set of 5 consecutive integers that are the product of a prime and another prime squared?

No Solution Yet Submitted by Math Man    
Rating: 5.0000 (1 votes)

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Hints/Tips Found Five Comment 8 of 8 |
I had a thought that another way to express this problem is to first think of the sequence of numbers of the form p*q^2: 12,18,20,28,44,45...  Then the problem asks us to find runs consecutive integers in this sequence.

The OEIS does have this as sequence A054753. They have a link to a stackexchange question which asks and eventually answers the problem.  The first set of five consecutive integers starts with 10093613546512321.

A030515 (numbers with exactly 6 factors) is a superset of A054753 and has some more information relevant to the problem. Specifically a reference to A141621 which is the sequence of the runs of five consecutive numbers in A030515/A054753.

  Posted by Brian Smith on 2017-04-02 21:17:20
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