277 (number)

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276 277 278
Cardinal two hundred seventy-seven
Ordinal 277th
(two hundred and seventy-seventh)
Factorization 277
Prime yes
Roman numeral CCLXXVII
Binary 1000101012
Ternary 1010213
Quaternary 101114
Quinary 21025
Senary 11416
Octal 4258
Duodecimal 1B112
Hexadecimal 11516
Vigesimal DH20
Base 36 7P36

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

Mathematical properties[edit]

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 + \frac{1}{2} x^2 + \frac{5}{24} x^4 + \frac{61}{720} x^6 + \frac{277}{8064} x^8 + \cdots

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]

References[edit]

  1. ^ "Sloane's A007703 : Regular primes", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  2. ^ "Sloane's A016088 : a(n) = smallest prime p such that Sum_{ primes q = 2, ..., p} 1/q exceeds n", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ "Sloane's A006450 : Primes with prime subscripts", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ Fernandez, Neil (1999), An order of primeness, F(p) .
  5. ^ "Sloane's A002585 : Largest prime factor of 1 + (product of first n primes)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ "Sloane's 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", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ "Sloane's A001608 : Perrin sequence (or Ondrej Such sequence): a(n) = a(n-2) + a(n-3)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. ^ Füredi, Z. (1987), "The number of maximal independent sets in connected graphs", Journal of Graph Theory 11 (4): 463–470, doi:10.1002/jgt.3190110403 .
  9. ^ "Sloane's A002478 : Bisection of A000930", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. ^ "Sloane's A051894 : Number of monic polynomials with integer coefficients of degree n with all roots in unit disc", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ "Sloane's A118312 : Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. ^ "Sloane's A046976 : Numerators of Taylor series for sec(x) = 1/cos(x)", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ "Sloane's A006378 : Prime self (or Colombian) numbers: primes not expressible as the sum of an integer and its digit sum", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.