Problems & Puzzles: Puzzles

 

Puzzle 845. 31 consecutive integers with an even quantity of prime factors

G. L. Honaker, Jr. in his Curio 31, wrote that:

"First occurrence of a run of exactly 31 consecutive integers with an even number of prime factors has never been found."

In the sequence A275508 we may see the first 30 first occurrences:

A275508 First occurrence of a run of exactly n consecutive integers with an even number of prime factors.

1, 9, 14, 33, 54, 201, 140, 2170, 213, 4529, 10403, 1934, 35811, 162144, 38405, 414225, 200938, 389409, 1792209, 5606457, 18493931, 30947532, 61190563, 8405437, 203631499, 577699919, 158280474, 68780189, 4082994208, 3944563444

As a matter of fact other larger occurrences are known. See:

A066793 The start of a record-breaking run of consecutive integers with an even number of prime factors

1, 9, 14, 33, 54, 140, 213, 1934, 35811, 38405, 200938, 389409, 1792209, 5606457, 8405437, 68780189, 880346227, 85910903650, 87635944146, 284340107954, 2195312273836, 2864568747364, 3155591650029.

I have calculated that:

880346227 is a run of 33 integers
85910903650  is a run of 34 integers
87635944146  is a run of 35 integers
284340107954  is a run of 37 integers
2195312273836  is a run of 38 integers
2864568747364  is a run of 41 integers
3155591650029  is a run of 44 integers

Accordingly with this, the sequence A275508 might be written this way, if question marks were admitted:

1, 9, 14, 33, 54, 201, 140, 2170, 213, 4529, 10403, 1934, 35811, 162144, 38405, 414225, 200938, 389409, 1792209, 5606457, 18493931, 30947532, 61190563, 8405437, 203631499, 577699919, 158280474, 68780189, 4082994208, 3944563444, ?, ?, 880346227, 85910903650, 87635944146, ?, 284340107954, 2195312273836, ?, ?, 2864568747364, ?, ?, 3155591650029, ...  

Q. Send your integers for exactly 31, 32, 36 39, 40, 42 or 43 consecutive integers with an even quantity of prime factors, or simply another record-breaking run larger than 44.


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