277 (number)

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

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

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]

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

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

1. ^ Sloane, N.J.A. (ed.). "Sequence A007703 (Regular primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. ^ Sloane, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence A002585 (Largest prime factor of 1 + (product of first n primes))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. ^ Sloane, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence A002478 (Bisection of A000930)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. ^ Sloane, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence 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, N.J.A. (ed.). "Sequence A046976 (Numerators of Taylor series for sec(x) = 1/cos(x))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
13. ^ Sloane, N.J.A. (ed.). "Sequence 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.