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%I #21 Feb 05 2019 07:52:05
%S 1,2,3,4,5,2,3,7,2,4,9,2,5,11,3,4,13,2,7,3,5,16,17,2,9,19,4,5,3,7,2,
%T 11,23,2,3,4,25,2,13,3,9,4,7,29,2,3,5,31,2,16,3,11,2,17,5,7,4,9,37,2,
%U 19,3,13,2,4,5,41,2,3,7,43,4,11,5,9,2,23,47,3,16,49,2,25
%N Triangle read by rows: n-th row contains Fermi-Dirac representation of n.
%C Unique factorization of n into distinct prime powers of form p^(2^k), cf. A050376: Product_{k=1..A064547(n)} T(n,k) = n.
%H Alois P. Heinz, <a href="/A213925/b213925.txt">Rows n = 1..8000, flattened</a> (first 1000 rows from Reinhard Zumkeller)
%H OEIS Wiki, <a href="/wiki/%22Fermi-Dirac_representation%22_of_n">"Fermi-Dirac representation" of n</a>
%e First rows:
%e . 1: 1
%e . 2: 2
%e . 3: 3
%e . 4: 4
%e . 5: 5
%e . 6: 2 3
%e . 7: 7
%e . 8: 2 4 8 = 2^2^0 * 2^2^1
%e . 9: 9
%e . 10: 2 5
%e .......
%e . 990: 2 5 9 11
%e . 991: 991
%e . 992: 2 16 31 992 = 2^2^0 * 2^2^2 * 31^2^0
%e . 993: 3 331
%e . 994: 2 7 71
%e . 995: 5 199
%e . 996: 3 4 83
%e . 997: 997
%e . 998: 2 499
%e . 999: 3 9 37 999 = 3^2^0 * 3^2^1 * 37^2^0
%e . 1000: 2 4 5 25 1000 = 2^2^0 * 2^2^1 * 5^2^0 * 5^2^1 .
%p T:= n-> `if`(n=1, [1], sort([seq((l-> seq(`if`(l[j]=1, i[1]^(2^(j-1)), [][]),
%p j=1..nops(l)))(convert(i[2], base, 2)), i=ifactors(n)[2])]))[]:
%p seq(T(n), n=1..60); # _Alois P. Heinz_, Feb 20 2018
%t nmax = 50; FDPrimes = Reap[k = 1; While[lim = nmax^(1/k); lim > 2, Sow[Prime[Range[PrimePi[lim]]]^k]; k = 2 k]][[2, 1]] // Flatten // Union;
%t f[1] = 1; f[n_] := Reap[m = n; Do[If[m == 1, Break[], If[Divisible[m, p], m = m/p; Sow[p]]], {p, Reverse[FDPrimes]}]][[2, 1]] // Reverse;
%t Array[f, nmax] // Flatten (* _Jean-François Alcover_, Feb 05 2019 *)
%o (Haskell)
%o a213925 n k = a213925_row n !! (k-1)
%o a213925_row 1 = [1]
%o a213925_row n = reverse $ fd n (reverse $ takeWhile (<= n) a050376_list)
%o where fd 1 _ = []
%o fd x (q:qs) = if m == 0 then q : fd x' qs else fd x qs
%o where (x',m) = divMod x q
%o a213925_tabf = map a213925_row [1..]
%Y Cf. A050376.
%Y For n > 1: A064547 (row lengths), A181894 (row sums), A223490, A223491.
%K nonn,tabf
%O 1,2
%A _Reinhard Zumkeller_, Mar 20 2013
%E Example corrected (row 992) by _Reinhard Zumkeller_, Mar 11 2015
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