A270599 - OEIS

%I #69 Oct 24 2018 08:20:36

%S 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,

%T 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,

%U 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

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

%C Number of partitions of n into such odd parts that the sum of their reciprocals is one. - _Antti Karttunen_, Jul 23 2018

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

%H David A. Corneth, <a href="/A270599/b270599.txt">Table of n, a(n) for n = 1..370</a> (terms 1..150 from Seiichi Manyama, terms 150..273 from Antti Karttunen)

%H David A. Corneth, <a href="/A270599/a270599_1.gp.txt">Tuples up to n = 370. </a>

%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>

%F a(2*k) = 0. - _David A. Corneth_, Jul 24 2018

%e 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.

%e 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

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

%o (Ruby)

%o def f(n)

%o   n - 1 + n % 2

%o end

%o def partition(n, min, max)

%o   return [[]] if n == 0

%o   [f(max), f(n)].min.step(min, -2).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

%o end

%o def A270599(n)

%o   ary = [1]

%o   (2..n).each{|m|

%o     cnt = 0

%o     partition(m, 2, m).each{|ary|

%o       cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1

%o     }

%o     ary << cnt

%o   }

%o   ary

%o end

%o (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

%o (PARI)

%o \\ More verbose version for computing values of a(n) for large n:

%o 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));

%o 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

%Y Cf. A000009, A051908.

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

%K nonn

%O 1,33

%A _Seiichi Manyama_, Mar 26 2016

%E Name corrected by _Antti Karttunen_, Jul 23 2018 at the suggestion of _David A. Corneth_