167 (number)

Last updated
166 167 168
Cardinal one hundred sixty-seven
Ordinal 167th
(one hundred sixty-seventh)
Factorization prime
Prime 39th, chen, gaussian, safe
Divisors 1, 167
Greek numeral ΡΞΖ´
Roman numeral CLXVII
Binary 101001112
Ternary 200123
Senary 4356
Octal 2478
Duodecimal 11B12
Hexadecimal A716

167 (one hundred [and] sixty-seven) is the natural number following 166 and preceding 168.

Contents

In mathematics

167 is the 39th prime number, an emirp, an isolated prime, a Chen prime, [1] a Gaussian prime, a safe prime, [2] and an Eisenstein prime with no imaginary part and a real part of the form .

167 is the smallest number which requires six terms when expressed using the greedy algorithm as a sum of squares, 167 = 144 + 16 + 4 + 1 + 1 + 1, [3] although by Lagrange's four-square theorem its non-greedy expression as a sum of squares can be shorter, e.g. 167 = 121 + 36 + 9 + 1.

167 is a full reptend prime in base 10, since the decimal expansion of 1/167 repeats the following 166 digits: 0.00598802395209580838323353293413173652694610778443113772455089820359281437125748502994 0119760479041916167664670658682634730538922155688622754491017964071856287425149700...

167 is a highly cototient number, as it is the smallest number k with exactly 15 solutions to the equation x - φ(x) = k. It is also a strictly non-palindromic number.

167 is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits, i. e., 1×6×7 = 3×(1+6+7)

167 is the smallest positive integer d such that the imaginary quadratic field Q(d) has class number = 11. [4]

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700 is the natural number following 699 and preceding 701.

600 is the natural number following 599 and preceding 601.

800 is the natural number following 799 and preceding 801.

145 is the natural number following 144 and preceding 146.

311 is the natural number following 310 and preceding 312.

129 is the natural number following 128 and preceding 130.

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152 is the natural number following 151 and preceding 153.

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References

  1. Sloane, N. J. A. (ed.). "SequenceA109611(Chen primes: primes p such that p + 2 is either a prime or a semiprime)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  2. Sloane, N. J. A. (ed.). "SequenceA005385(Safe primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  3. Sloane, N. J. A. (ed.). "SequenceA006892(Representation as a sum of squares requires n squares with greedy algorithm)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  4. "Tables of imaginary quadratic fields with small class number". numbertheory.org.