A213925 Triangle read by rows: n-th row contains Fermi-Dirac representation of n.

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, 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, 19, 3, 13, 2, 4, 5, 41, 2, 3, 7, 43, 4, 11, 5, 9, 2, 23, 47, 3, 16, 49, 2, 25

Offset: 1

Author

Reinhard Zumkeller, Mar 20 2013

Keywords

• nonn
• tabf

Comments

Unique factorization of n into distinct prime powers of form p^(2^k), cf. A050376.

Examples

			First rows:
.     1:    1
.     2:    2
.     3:    3
.     4:    4
.     5:    5
.     6:    2  3
.     7:    7
.     8:    2  4                   8 = 2^2^0 * 2^2^1
.     9:    9
.    10:    2  5
.......
.   990:    2   5  9  11
.   991:  991
.   992:    2  16 31             992 = 2^2^0 * 2^2^2 * 31^2^0
.   993:    3 331
.   994:    2   7 71
.   995:    5 199
.   996:    3   4 83
.   997:  997
.   998:    2 499
.   999:    3   9 37             999 = 3^2^0 * 3^2^1 * 37^2^0
.  1000:    2   4  5  25        1000 = 2^2^0 * 2^2^1 * 5^2^0 * 5^2^1 .
		

Links

• Alois P. Heinz, Rows n = 1..8000, flattened (first 1000 rows from Reinhard Zumkeller)
• OEIS Wiki, "Fermi-Dirac representation" of n.

Crossrefs

Cf. A050376.

For n > 1: A064547 (row lengths), A181894 (row sums), A223490, A223491.

Programs

Haskell

a213925 n k = a213925_row n !! (k-1)
a213925_row 1 = [1]
a213925_row n = reverse $ fd n (reverse $ takeWhile (<= n) a050376_list)
   where fd 1 _      = []
         fd x (q:qs) = if m == 0 then q : fd x' qs else fd x qs
                       where (x',m) = divMod x q
a213925_tabf = map a213925_row [1..]

Maple

T:= n-> `if`(n=1, [1], sort([seq((l-> seq(`if`(l[j]=1, i[1]^(2^(j-1)), [][]),
             j=1..nops(l)))(convert(i[2], base, 2)), i=ifactors(n)[2])]))[]:
seq(T(n), n=1..60);  # Alois P. Heinz, Feb 20 2018

Mathematica

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;
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;
Array[f, nmax] // Flatten (* Jean-François Alcover, Feb 05 2019 *)

PARI

row(n) = if(n == 1, [1], my(f = factor(n), p = f[, 1], e = f[, 2], r = [], b); for(i = 1, #p, b = binary(e[i]); for(j = 0, #b-1, if(b[#b-j], r = concat(r, p[i]^(2^j))))); r); \\ Amiram Eldar, May 02 2025

Formula

Product_{k=1..A064547(n)} T(n,k) = n.

Extensions

Example corrected (row 992) by Reinhard Zumkeller, Mar 11 2015