Stein, Ulam and Wells, "A visual display of some properties of the distribution of primes", American Mathematics Monthly 71 (5) 516-520.

A.K. Dewdney, "How to pan for primes in numerical gravel", Scientific American, July 1988, p.90-93.

S.M. Ellerstein, "The Pronic Renaissance: The Ulam Square Spiral (Modified)" Journal of Recreational Mathematics 29 (3), 1998, p. 188-189.

S.M. Ellerstein, "The Pronic Renaissance II: The Ellerstein Square Spiral", Journal of Recreational Mathematics 30 (4), 1999-2000, p. 246-250.
 

According to E. Weisstein's Mathworld site,

"Remarkably, noted science fiction author Arthur C. Clarke described the prime spiral in his novel The City and the Stars (1956, Ch. 6, p. 54). Clarke wrote, "Jeserac sat motionless within a whirlpool of numbers. The first thousand primes.... Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed. He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh."

However, Clarke never actually performed this thought experiment (pers. comm. to E. Pegg Jr., May 27, 2002), thus leaving discovery of the unexpected properties of the prime spiral to Ulam."

More about this, including an explanatory comment from Clarke himself can be found here.