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%I #28 May 09 2021 02:16:11
%S 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,
%T 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,
%U 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
%N Table for the unique factorization of integers >= 2 into terms of A186285 or their squares.
%C 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.
%C 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).
%C 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, ...].
%H Michael De Vlieger, <a href="/A240230/b240230.txt">Table of n, a(n) for n = 1..13622</a> (Rows 1 <= n <= 5000).
%e The irregular triangle a(n,k) starts (in the first part the factors are listed):
%e n\k 1 2 3 ... multiplicity sequence
%e 1: 1 0-sequence [repeat(0,)]
%e 2: 2 [1]
%e 3: 3 [0, 1]
%e 4: 2, 2 [2]
%e 5: 5 [0, 0, 1]
%e 6: 2, 3 [1, 1]
%e 7: 7 [0, 0, 0, 1]
%e 8: 8 [0, 0, 0, 0, 1]
%e 9: 3, 3 [0, 2]
%e 10: 2, 5 [1, 0, 1]
%e 11: 11 [0, 0, 0, 0, 0, 1]
%e 12: 2, 2, 3 [2, 1]
%e 13: 13 [0, 0, 0, 0, 0, 0, 1]
%e 14: 2, 7 [1, 0, 0, 1]
%e 15: 3, 5 [0, 1, 1]
%e 16: 2, 8 [1, 0, 0, 0, 1]
%e 17: 17 [0, 0, 0, 0, 0, 0, 0, 1]
%e 18: 2, 3, 3 [1, 2]
%e 19: 19 [0, 0, 0, 0, 0, 0, 0, 0, 1]
%e 20: 2, 2, 5 [2, 0, 1]
%e ...(reformatted - _Wolfdieter Lang_, May 16 2014)
%t 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 *)
%Y Cf. A050376, A186285, A213925, A240231.
%K nonn,tabf
%O 1,2
%A _Wolfdieter Lang_, May 15 2014
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