A256221 Number of distinct nonzero Fibonacci numbers in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.
1, 2, 3, 4, 5, 6, 8, 8, 8, 12, 12, 13, 13, 13, 13, 15, 15, 15, 17, 17, 17, 19, 21, 21, 23, 24, 25, 25, 25, 25, 25, 27
Offset: 1
Examples
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.
Programs
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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]]] -
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
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
Comments