A240230 Table for the unique factorization of integers >= 2 into terms of A186285 or their squares.

1, 2, 3, 2, 2, 5, 2, 3, 7, 8, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 8, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 3, 8, 5, 5, 2, 13, 27, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 8, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 5, 8, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5, 2, 23, 47, 2, 3, 8, 7, 7, 2, 5, 5

Offset: 1

Author

Wolfdieter Lang, May 15 2014

Keywords

• nonn
• tabf

Comments

The terms of A186285 are primes to powers of 3 (PtPP(p=3) primes to prime powers with p=3). See A050376 for PtPP(2), appearing in the OEIS as 'Fermi-Dirac' primes, because in this case the unique representation of n >= 2 works with distinct members of A050376, hence the multiplicity (occupation number) is either 0 (not present) or 1 (appearing once). For p=3 the multiplicities are 0, 1, 2. See the multiplicity sequences given in the examples. At position m the multiplicity for A186285(m), m >= 1, is recorded, and trailing zeros are omitted, except for n = 1.

In order to include n=1 one defines as its representation 1, even though 1 is not a member of A186285 (in order to have a unique representation for n >= 2 modulo commutation of factors).

The length of row n (the number of factors) is obtained from the (reversed) base 3 representation of the exponents of the primes appearing in the ordinary factorization of n, by adding all entries. E.g., n = 2^5*5^7 = 2500000 will have row length 6 because (5)(3r) = [2, 1] and (7)(3r) = [1, 2] (reversed base 3), leading to the 6 factors (2^2*8^1)*(5^1*125^2) = 2*2*5*8*125*125. The row length sequence is A240231 = [1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, ...].

Examples

			The irregular triangle a(n,k) starts (in the first part the factors are listed):
  n\k   1  2  3 ...     multiplicity sequence
  1:    1               0-sequence [repeat(0,)]
  2:    2               [1]
  3:    3               [0, 1]
  4:    2, 2            [2]
  5:    5               [0, 0, 1]
  6:    2, 3            [1, 1]
  7:    7               [0, 0, 0, 1]
  8:    8               [0, 0, 0, 0, 1]
  9:    3, 3            [0, 2]
  10:   2, 5            [1, 0, 1]
  11:  11               [0, 0, 0, 0, 0, 1]
  12:   2, 2, 3         [2, 1]
  13:  13               [0, 0, 0, 0, 0, 0, 1]
  14:   2, 7            [1, 0, 0, 1]
  15:   3, 5            [0, 1, 1]
  16:   2, 8            [1, 0, 0, 0, 1]
  17:  17               [0, 0, 0, 0, 0, 0, 0, 1]
  18:   2, 3, 3         [1, 2]
  19:  19               [0, 0, 0, 0, 0, 0, 0, 0, 1]
  20:   2, 2, 5         [2, 0, 1]
...(reformatted - _Wolfdieter Lang_, May 16 2014)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..13622 (Rows 1 <= n <= 5000).

Crossrefs

Cf. A050376, A186285, A213925, A240231.

Programs

Mathematica

With[{s = Select[Select[Range[53], PrimePowerQ], IntegerQ@Log[3, FactorInteger[#][[1, -1]]] &]}, {{1}}~Join~Table[Reverse@ Rest@ NestWhileList[Function[{k, m}, {k/#, #} &@ SelectFirst[Reverse@ TakeWhile[s, # <= k &], Divisible[k, #] &]] @@ # &, {n, 1}, First@ # > 1 &][[All, -1]], {n, 2, Max@ s}]] // Flatten (* Michael De Vlieger, Aug 14 2017 *)