Explain the pattern and find the next five rows.

Hint: the symbols I chose have nothing to do with the pattern.

For every integer n, there is an integer f which is the product of the prime factors of n.  The nth element is f symbols long.  The kth symbol in an element is '+' if the gcd of f and k is 1, otherwise the symbol is '='.

Example 1: n=12
12 = 2*2*3, then f = 2*3 = 6
gcd(6,1) = 1 '+'
gcd(6,2) = 2 '='
gcd(6,3) = 3 '='
gcd(6,4) = 2 '='
gcd(6,5) = 1 '+'
gcd(6,6) = 6 '='
6: +===+=

Example 2: n=10
10 = 2*5, then f = 2*5 = 10
gcd(10,1) = 1 '+'
gcd(10,2) = 2 '='
gcd(10,3) = 1 '+'
gcd(10,4) = 2 '='
gcd(10,5) = 5 '='
gcd(10,6) = 2 '='
gcd(10,7) = 1 '+'
gcd(10,8) = 2 '='
gcd(10,9) = 1 '+'
gcd(10,10) = 10 '='
10: +=+===+=+=

Continuing this algorithm for n=13 to 20 yeilds:
13: ++++++++++++=
14: +=+=+===+=+=+=
15: ++=+==++==+=++=
16: +=
17: ++++++++++++++++=
18: +===+=
19: ++++++++++++++++++=
20: +=+===+=+=

Brian Wainscott asked what would happen for n=30: +=====+===+=+===+=+===+=====+=