Ulam wrote the positive integers in a rectilinear spiral pattern and observed that primes showed up on diagonals rather more than they might have been expected to. Here are images illustrating the same kind of thing with other sequences of positive integers. Since there seems to be no convention on this sort of thing, my spirals start with 1, and then move upward to 2, then clockwise, like this.
primes
twin primes (the lower twin)
squares
triangular numbers
pentagonal numbers
abundant numbers
abundant numbers (low res)
tau(n) divides n
tau(n) equals tau(n+1)
tau(n) equals tau(n+2)
tau(n), greyscale
happy numbers
modulo 2 (low res)
modulo 3 (low res)
modulo 4 (low res)
modulo 5 (low res)
modulo 8 (low res)
modulo 11 (low res)
modulo 13 (low res)
modulo 29 
modulo 30
modulo 30, greyscale
modulo 31