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

1,2

Comments

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

Links

Alois P. Heinz, Rows n = 1..8000, flattened (first 1000 rows from Reinhard Zumkeller)

OEIS Wiki, "Fermi-Dirac representation" of n.

Formula

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

Example

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 .

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 *)

Prog

(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..]
(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

Crossrefs

Cf. A050376.

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

Sequence in context: A329243 A375636 A161768 * A394674 A141810 A141809

Adjacent sequences: A213922 A213923 A213924 * A213926 A213927 A213928

Keyword

nonn,tabf

Author

Reinhard Zumkeller, Mar 20 2013

Extensions

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

Status

approved