A270599 Number of ways to express 1 as the sum of unit fractions with odd denominators such that the sum of those denominators is n.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0

Offset

1,33

Comments

Number of partitions of n into such odd parts that the sum of their reciprocals is one. - Antti Karttunen, Jul 23 2018

It would be nice to know whether nonzero values may occur only on n of the form 8k+1.

Links

David A. Corneth, Table of n, a(n) for n = 1..370 (terms 1..150 from Seiichi Manyama, terms 150..273 from Antti Karttunen)

David A. Corneth, Tuples up to n = 370.

Index entries for sequences related to Egyptian fractions

Formula

a(2*k) = 0. - David A. Corneth, Jul 24 2018

Example

1 = 1/3 + 1/3 + 1/3, the sum of denominators is 9, this is the only expression of 1 as unit fractions with odd denominators that sum to 9, so a(9)=1.
1 = 1/15 + 1/5 + 1/5 + 1/5 + 1/3 = 1/9 + 1/9 + 1/9 + 1/3 + 1/3 are the only solutions with odd denominators that sum to 33, thus a(33) = 2. - Antti Karttunen, Jul 24 2018

Mathematica

Array[Count[IntegerPartitions[#, All, Range[1, #, 2]], _?(Total[1/#] == 1 &)] &, 70] (* Michael De Vlieger, Jul 26 2018 *)

Prog

(Ruby)
def f(n)
  n - 1 + n % 2
end
def partition(n, min, max)
  return [[]] if n == 0
  [f(max), f(n)].min.step(min, -2).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A270599(n)
  ary = [1]
  (2..n).each{|m|
    cnt = 0
    partition(m, 2, m).each{|ary|
      cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
    }
    ary << cnt
  }
  ary
end
(PARI) A270599(n, maxfrom=n, fracsum=0) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 23 2018
(PARI)
\\ More verbose version for computing values of a(n) for large n:
A270599(n) = if(!(n%2), 0, my(s=0); forstep(k = n, 1, -2, print("A270599(", n, ") at toplevel, k=", k, " s=", s); s += A270599aux(n-k, min(k, n-k), 1/k)); (s));
A270599aux(n, maxfrom, fracsum) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599aux(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 24 2018

Crossrefs

Cf. A000009, A051908.

Cf. also A201644, A201646, A201647, A201648, A201649.

Sequence in context: A331984 A087781 A181009 * A091398 A062103 A112314

Adjacent sequences: A270596 A270597 A270598 * A270600 A270601 A270602

Keyword

nonn

Author

Seiichi Manyama, Mar 26 2016

Extensions

Name corrected by Antti Karttunen, Jul 23 2018 at the suggestion of David A. Corneth

Status

approved