1613

This number is a prime.

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1613^2 = 867253 + 867257 + 867259 (3 consecutive primes) = 25247 + 25253 + ... + 26251 + 26261 (101 consecutive primes).

1613 is a prime such that the previous two primes AND the next two primes are twin primes.

There are 1613 prime sextuplets (p, p+4, p+6, p+10, p+12, p+16 all prime) < 10^10.

1613 is the 17th prime in the prime sequence f(n)=4*n^2 - 4*n + 653 (note that f(0), f(1), ... , f(16), f(17), f(18) are all primes, while f(19) is not).

(760*10^1613 - 31)/9 = 8 (4)1613 1 is prime (note the 1613 consecutive 4's in the middle).

The largest prime in the first 4-digit cousin prime pair (p, q) such that p^2+q^3 -/+1 is a twin prime pair.

82^1613 - 81^1613 is prime.

1613 is the largest prime factor of 2^26+1.

π(1613) = M(8); π(1613) + 2*1613 is a perfect square of a prime (59^2).

(In case it is not obvious, 1613 is one of my favorite small 4-digit primes not related to the Mersenne primes. :-)) [Noll]