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[edit] Largest known Ruth-Aaron pair

Let p = 4941451234 × 2411# + 1 (a prime found by Markus Frind and Paul Underwood), and n = p × ((p − 1) × (p − 145735591) − 1).
(n, n + 1) is a 3109-digit Ruth-Aaron pair, found by Jens Kruse Andersen and the largest known as of May 2006. [1]

Alternatively (avoiding computation from formula in source):

As of May 2006, the largest known Ruth-Aaron pair has 3109 digits, found by Jens Kruse Andersen. [2]

I suggest one of the above to be added to the article. I am Jens Kruse Andersen, so I suggest it here per WP:COI. PrimeHunter 14:29, 6 February 2007 (UTC)

[edit] Parser test

4*6/3+8-1 = 15

[edit] Dihedral prime

A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside down), and surface (actual display or reflection on a mirror). The first few base 10 dihedral primes are

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (sequence A038136 in OEIS)[1]

The smallest dihedral prime that reads differently with each orientation and surface combination is 120121 which becomes 150151 (upside down), 151051 (mirrored), and 121021 (both upside down and mirrored).

The digits 0, 1 and 8 remain the same regardless of orientation or surface (the fact that 1 moves from the right to the left of the seven-segment cell when reversed is ignored). 2 and 5 turn into each other when viewed upside down or reflected in a mirror, so they turn back into themselves when both upside down and mirrored. In the display of a calculator that can handle hexadecimal, 3 would become E reflected, but E being an even digit, the 3 can't be used as the first digit because the reflected number will be even. Though 6 and 9 become each other upside down, they are not valid digits when reflected, at least not in any of the numeral systems pocket calculators usually operate in.

Strobogrammatic primes that don't use 6 or 9 are dihedral primes. This includes repunit primes and all other palindromic primes which only contain digits 0, 1 and 8 (in binary, all palindromic primes are dihedral). It appears to be unknown whether there exist infinitely many dihedral primes, but this would follow from the conjecture that there are infinitely many repunit primes.

[edit] References

  1. ^ A038136 is missing the dihedral prime 5. Retrieved on March 4, 2007.
  • Eric W. Weisstein. Dihedral Prime. MathWorld – A Wolfram Web Resource.


Moebius function

[edit] Properties

Many properties of natural numbers can be seen or directly computed from their prime factorization.

  • The multiplicity of a prime factor p of n is the largest exponent m for which pm divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
  • Ω(n), the big Omega function, is the number of prime factors of n counted with multiplicity (so it's the sum of all prime factor multiplicities).
  • A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11. There are many special types of prime numbers.
  • A composite number has Ω(n) > 1. The first: 4, 6, 8, 9, 10. All numbers above 1 are either prime or composite. 1 is neither.
  • A semiprime has Ω(n) = 2 (so it's composite). The first: 4, 6, 9, 10, 14.
  • A k-almost prime (for a natural number k) has Ω(n) = k (so it's composite if k > 1).
  • An even number has the prime factor 2. The first: 2, 4, 6, 8, 10.
  • An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9. All integers are either even or odd.
  • A square has even multiplicity for all prime factors (it's of the form a2 for some a). The first: 1, 4, 9, 16, 25.
  • A cube has all multiplicities divisible by 3 (it's of the form a3 for some a). The first: 1, 8, 27, 64, 125.
  • A perfect power has a common divisor m > 1 for all multiplicities (it's of the form am for some a > 1 and m > 1).
  • A square-free integer has no prime factor with multiplicity above 1. The first: 1, 2, 3, 5, 6.
  • A squareful number has multiplicity above 1 for all prime factors. A number where some but not all prime factors have multiplicity above 1 is neither squarefree nor squareful.
  • A sphenic number has Ω(n) = 3 and is square free (so it's the product of 3 distinct primes). The first: 30, 42, 66, 70, 78.
  • a0(n) is the sum of primes dividing n, counted with multiplicity. It is an additive function.
  • A Ruth-Aaron pair is two consecutive numbers (x, x+1) with a0(x) = a0(x+1).
  • A primorial x# is the product of all primes from 2 to x. The first: 2, 6, 30, 210, 2310 (1# = 1 may be included).
  • A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120.
  • A k-smooth number (for a natural number k) has largest prime factor ≤ k (so it's also j-smooth for any j > k).
  • m is smoother than n if the largest prime factor of m is below the largest of n.
  • gcd(m, n) (greatest common divisor of m and n) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n).
  • m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor).
  • lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n).
  • gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than to compute gcd and lcm with other algorithms which do not require known prime factorization.
  • m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n.

The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.

[edit] Blacklist test

Blacklist test with random blacklisted link.

http://automotiveoilchange.net

automotiveoilchange.net

http://www.onelegout.com/stencil_tutorial.html