Problems & Puzzles: Problems Problem 28. K least consecutive numbers n such that are not squarefree The first (least) occurrence of K=3 consecutive numbers that are not squarefree starts with the number 48, because 48 = 2^{4}.3, 49 = 7^{2} & 50 = 2.5^{2}. This kind of sequences may also be named as "gaps of square free consecutive numbers" and also as "sequences of consecutive numbers whose Möbius function is zero". Here is the list of the sequences found up to November 1999 for K=1 to 15.
This table corresponds to the sequence A051681 of the Sloane's OnLine Encyclopedia of Integer Sequences. The first eleven terms were sent by Erich Friedmann and Patrick De Geest, while the last four terms were found by Louis Marmet (louis at marmet.org) and David Bernier (ezcos@yahoo.com) very recently (November 1999). The entries for K=16 & 17 were reported to me by J. K. Andersen (October 2003) as the last terms of the sequence A045882 Marmet and (independently) Bernier found also that "No gaps longer than 15 were found up to N = 96×10^{12}^{ }" and I would strongly recommend to consider an Algorithm developed by Marmet before going farther. This algorithm is " based on the sieve of Eratosthenes" and " is much faster" than the algorithm of "Mathematica", according to Marmet. Questions:
SOLUTION Louis Marmet has sent a methodic way of finding one solution for a sequence 16 members of this type. As fas as I understood his answer for the question 2 of this puzzle, Marmet also solves the question 1. Do you agree? Here is his email: "You
were asking if I could find a solution for K=16 using the method. I
don't ...For
gaps with K=16, this set of 12 congruences would do it: It
is easier to see here what is happening. Note that because N+16 = 0 (mod
4), What I (CR) have done is simply to program in Ubasic a code to solve this set of 12 modular equations sent by Marmet to get a set of 17 consecutive numbers that are not square free. My code pinted out the following result for the initial member of that set, N: N=316524040719565110079148 *** Louise Marmet wrote:
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