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613 = Primes: prime number, first number of prime triple ( p, p + 4, p + 6 ), middle number of sexy prime triple ( p-6, p, p + 6 ).

Geometrical numbers: Centered square number with 18 per side, circular number of 21 with a square grid and 27 using a triangular grid.

Also 17-gonal.

Hypotenuse of a right triangle with integral sides, these being 35 and 612.

Partitioning: 613 partitions of 47 into non-factor primes, 613 non-squashing partitions into distinct parts of the number 54.

Squares: Sum of squares of two consecutive integers, 17 and 18.

Additional properties: a lucky number.