OFFSET
1,2
COMMENTS
EXAMPLE
a(4) = 4 because 4 sums yield distinct Fibonacci numerators: 1, 1 + 1/2 = 3/2, 1/2 + 1/3 = 5/6 and 1/2 + 1/3 + 1/4 = 13/12.
MAPLE
S:= {0, 1}: N:= {1}:
nfibs:= 10:
fibs:= {seq(combinat:-fibonacci(n), n=1..nfibs)}:
A[1]:= 1:
fibnums:= {1}:
for n from 2 to 24 do
Sp:= map(`+`, S, 1/n);
N:= N union map(numer, Sp);
Nmax:= max(N);
S:= S union Sp;
while combinat:-fibonacci(nfibs) < Nmax do nfibs:= nfibs+1; fibs:= fibs union {combinat:-fibonacci(nfibs)} od;
newfibnums:= N intersect fibs;
fibnums:= newfibnums;
A[n]:= nops(fibnums);
od:
seq(A[n], n=1..24); # Robert Israel, Dec 09 2016
MATHEMATICA
<<"DiscreteMath`Combinatorica`"; maxN=23; For[prms={}; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], prms=Union[prms, {k}]]]; Print[Length[prms]]]
PROG
(Python)
from math import gcd, lcm
from itertools import combinations
def A256221(n):
m = lcm(*range(1, n+1))
fset, fibset, mlist = set(), set(), tuple(m//i for i in range(1, n+1))
a, b, k = 0, 1, sum(mlist)
while b <= k:
fibset.add(b)
a, b = b, a+b
for l in range(1, n//2+1):
for p in combinations(mlist, l):
s = sum(p)
if (t := s//gcd(s, m)) in fibset:
fset.add(t)
if 2*l != n and (t := (k-s)//gcd(k-s, m)) in fibset:
fset.add(t)
if (t:= k//gcd(k, m)) in fibset: fset.add(t)
return len(fset) # Chai Wah Wu, Feb 15 2022
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Mar 19 2015
EXTENSIONS
Corrected and more terms added by Robert Israel, Dec 09 2016
a(29)-a(31) from Chai Wah Wu, Feb 15 2022
a(32) from Chai Wah Wu, Feb 16 2022
STATUS
approved