oeis-series

WIP

Demo site: https://aborg0.github.io/oeis-series/

Development

Use the sbt task oeisSeriesJS/fastOptJS::webpack to generate the necessary js files

Usage

Try the following:

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 0)

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 1)

  • left(m) := { m < 2: betweenAdjacentSameDenominatorsCountedOnce(m, 1); : betweenAdjacentSameDenominatorsCountedOnce(m, 1) - betweenAdjacentSameDenominatorsCountedOnce(m - 2, 1)}

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 2)

  • left(m) := { m < 6: betweenAdjacentSameDenominatorsCountedOnce(m, 2); : betweenAdjacentSameDenominatorsCountedOnce(m, 2) - betweenAdjacentSameDenominatorsCountedOnce(m - 6, 2)}

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 3)

  • left(m) := { m < 12: betweenAdjacentSameDenominatorsCountedOnce(m, 3); : betweenAdjacentSameDenominatorsCountedOnce(m, 3) - betweenAdjacentSameDenominatorsCountedOnce(m - 12, 3)}

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 4)

  • left(m) := { m < 20: betweenAdjacentSameDenominatorsCountedOnce(m, 4); : betweenAdjacentSameDenominatorsCountedOnce(m, 4) - betweenAdjacentSameDenominatorsCountedOnce(m - 20, 4)}

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 5)

  • left(m) := { m < 30: betweenAdjacentSameDenominatorsCountedOnce(m, 5); : betweenAdjacentSameDenominatorsCountedOnce(m, 5) - betweenAdjacentSameDenominatorsCountedOnce(m - 30, 5)}

• left(m) := betweenAdjacentSameDenominatorsCountedOnce(m, 6)

  • left(m) := { m < 42: betweenAdjacentSameDenominatorsCountedOnce(m, 6); : betweenAdjacentSameDenominatorsCountedOnce(m, 6) - betweenAdjacentSameDenominatorsCountedOnce(m - 42, 6)}

• ... Note that 2 = 1*2, 6 = 2*3, 12 = 3*4, 20 = 4*5, 30 = 5*6, 42 = 6*7, ...

• let's see from the middle to the left/right (it is symmetric)

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 + 1, m)

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2, m)

  • middle(m) := { m < 2: betweenAdjacentSameDenominatorsCountedOnce(m * 2, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 2) * 2, m - 2)}

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 1, m)

  • middle(m) := { m <= 6: betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 1, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 1, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 6) * 2 - 1, m - 6)}

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 2, m)

  • middle(m) := { m <= 12: betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 2, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 2, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 12) * 2 - 2, m - 12)}

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 3, m)

  • middle(m) := { m <= 32: betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 3, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 3, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 30) * 2 - 3, m - 30)}

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 4, m)

  • middle(m) := { m <= 65: betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 4, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 4, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 60) * 2 - 4, m - 60)}

• middle(m) := betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 5, m)

  • middle(m) := { m <= 212: betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 5, m); : betweenAdjacentSameDenominatorsCountedOnce(m * 2 - 5, m) - betweenAdjacentSameDenominatorsCountedOnce((m - 210) * 2 - 5, m - 210)}

• ... Note that 2, 6, 12, 30, 60, 210, ... might suggest that the next pattern length might be 420.

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 0)

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 1)

  • left(m) := { m <= 2: betweenAdjacentSameDenominatorsCountedEach(m, 1); : betweenAdjacentSameDenominatorsCountedEach(m, 1) - betweenAdjacentSameDenominatorsCountedEach(m - 2, 1)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 2)

  • left(m) := { m <= 6: betweenAdjacentSameDenominatorsCountedEach(m, 2); : betweenAdjacentSameDenominatorsCountedEach(m, 2) - betweenAdjacentSameDenominatorsCountedEach(m - 6, 2)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 3)

  • left(m) := { m <= 12: betweenAdjacentSameDenominatorsCountedEach(m, 3); : betweenAdjacentSameDenominatorsCountedEach(m, 3) - betweenAdjacentSameDenominatorsCountedEach(m - 12, 3)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 4)

  • left(m) := { m <= 20: betweenAdjacentSameDenominatorsCountedEach(m, 4); : betweenAdjacentSameDenominatorsCountedEach(m, 4) - betweenAdjacentSameDenominatorsCountedEach(m - 20, 4)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 5)

  • left(m) := { m <= 30: betweenAdjacentSameDenominatorsCountedEach(m, 5); : betweenAdjacentSameDenominatorsCountedEach(m, 5) - betweenAdjacentSameDenominatorsCountedEach(m - 30, 5)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 6)

  • left(m) := { m <= 42: betweenAdjacentSameDenominatorsCountedEach(m, 6); : betweenAdjacentSameDenominatorsCountedEach(m, 6) - betweenAdjacentSameDenominatorsCountedEach(m - 42, 6)}

• left(m) := betweenAdjacentSameDenominatorsCountedEach(m, 7)

  • left(m) := { m <= 56: betweenAdjacentSameDenominatorsCountedEach(m, 7); : betweenAdjacentSameDenominatorsCountedEach(m, 7) - betweenAdjacentSameDenominatorsCountedEach(m - 56, 7)}

• ... Note that 2 = 1*2, 6 = 2*3, 12 = 3*4, 20 = 4*5, 30 = 5*6, 42 = 6*7, 56 = 7*8, ...

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 + 1, m)

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2, m)

  • middle(m) := { m < 2: betweenAdjacentSameDenominatorsCountedEach(m * 2, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2, m) - betweenAdjacentSameDenominatorsCountedEach((m - 2) * 2, m - 2)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 1, m)

  • middle(m) := { m <= 5: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 1, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 1, m) - betweenAdjacentSameDenominatorsCountedEach((m - 3) * 2 - 1, m - 3)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 2, m)

  • middle(m) := { m <= 11: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 2, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 2, m) - betweenAdjacentSameDenominatorsCountedEach((m - 8) * 2 - 2, m - 8)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 3, m)

  • middle(m) := { m <= 19: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 3, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 3, m) - betweenAdjacentSameDenominatorsCountedEach((m - 15) * 2 - 3, m - 15)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 4, m)

  • middle(m) := { m <= 29: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 4, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 4, m) - betweenAdjacentSameDenominatorsCountedEach((m - 24) * 2 - 4, m - 24)}
  • middle(m) := { m <= 17: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 4, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 4, m) - betweenAdjacentSameDenominatorsCountedEach((m - 12) * 2 - 4, m - 12)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 5, m)

  • middle(m) := { m <= 38: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 5, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 5, m) - betweenAdjacentSameDenominatorsCountedEach((m - 35) * 2 - 5, m - 35)}

• middle(m) := betweenAdjacentSameDenominatorsCountedEach(m * 2 - 6, m)

  • middle(m) := { m <= 52: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 6, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 6, m) - betweenAdjacentSameDenominatorsCountedEach((m - 48) * 2 - 6, m - 48)}
  • middle(m) := { m <= 28: betweenAdjacentSameDenominatorsCountedEach(m * 2 - 6, m); : betweenAdjacentSameDenominatorsCountedEach(m * 2 - 6, m) - betweenAdjacentSameDenominatorsCountedEach((m - 24) * 2 - 6, m - 24)}

• ... Note that 3 = 1*3, 8 = 2*4, 15 * 3*5, 24 = 4*6, 35 = 5*7, 48 = 6*8, ...

• Check the rows for primes of the betweenAdjacentSameDenominatorsCountedEach:

  • same(m) := betweenAdjacentSameDenominatorsCountedEach(3, m)
    • same(m) := betweenAdjacentSameDenominatorsCountedEach(4, m)!
  • same(m) := betweenAdjacentSameDenominatorsCountedEach(5, m)
  • same(m) := betweenAdjacentSameDenominatorsCountedEach(7, m)
  • same(m) := betweenAdjacentSameDenominatorsCountedEach(11, m)
  • same(m) := betweenAdjacentSameDenominatorsCountedEach(13, m)
  • same(m) := betweenAdjacentSameDenominatorsCountedEach(17, m)
  • ... (It is not too hard to prove that this is always the same (p-1)/2 for odd prime p, except for the number of fractions (with denominator less than p) between 0 and 1/p and (p-1)/p and 1 as there are 0.)

• Let's see some Carmichael numbers too:

  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(561, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(1105, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(1729, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(2465, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(2821, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(6601, m)
  • notsame(m) := betweenAdjacentSameDenominatorsCountedEach(8911, m)