277 (number) explained

277 (two hundred [and] seventy-seven) is the natural number following 276 and preceding 278.

Number:	277
Prime:	yes

Mathematical properties

277 is the 59th prime number, and is a regular prime.^[1] It is the smallest prime p such that the sum of the inverses of the primes up to p is greater than two.^[2] Since 59 is itself prime, 277 is a super-prime.^[3] 59 is also a super-prime (it is the 17th prime), as is 17 (the 7th prime). However, 7 is the fourth prime number, and 4 is not prime. Thus, 277 is a super-super-super-prime but not a super-super-super-super-prime.^[4] It is the largest prime factor of the Euclid number 510511 = 2 × 3 × 5 × 7 × 11 × 13 × 17 + 1.^[5]

As a member of the lazy caterer's sequence, 277 counts the maximum number of pieces obtained by slicing a pancake with 23 straight cuts.^[6] 277 is also a Perrin number, and as such counts the number of maximal independent sets in an icosagon.^{[7] [8]} There are 277 ways to tile a 3 × 8 rectangle with integer-sided squares,^[9] and 277 degree-7 monic polynomials with integer coefficients and all roots in the unit disk.^[10] On an infinite chessboard, there are 277 squares that a knight can reach from a given starting position in exactly six moves.^[11]

277 appears as the numerator of the fifth term of the Taylor series for the secant function:^[12]

\sec x = 1 + (1)/(2) x ^{2} + (5)/(24) x ^{4} + (61)/(720) x ^{6} + (277)/(8064) x ^{8} + …

Since no number added to the sum of its digits generates 277, it is a self number. The next prime self number is not reached until 367.^[13]

Notes and References

1. A007703. Regular primes.
2. A016088. a(n) = smallest prime p such that Sum_ 1/q exceeds n.
3. A006450. Primes with prime subscripts.
4. .
5. A002585. Largest prime factor of 1 + (product of first n primes).
6. A000124. Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.
7. A001608. Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3).
8. .
9. A002478. Bisection of A000930.
10. A051894. Number of monic polynomials with integer coefficients of degree n with all roots in unit disc.
11. A118312. Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square.
12. A046976. Numerators of Taylor series for sec(x) = 1/cos(x).
13. A006378. Prime self (or Colombian) numbers: primes not expressible as the sum of an integer and its digit sum.