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 A256220 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number. 4
 1, 3, 5, 9, 11, 22, 28, 37, 45, 62, 70, 125, 133, 172, 330, 421, 450, 840, 901, 1710, 2356, 2724, 2824, 5367, 6022, 7142, 8072, 18771, 19204, 35739, 36453, 42853, 82094, 88574, 155642, 264869 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Note that for each n there are only 2^(n-1) new sums to consider. For the number of distinct Fibonacci numbers, see A256221. For the largest generated Fibonacci number, see A256222. For the smallest Fibonacci number not generated, see A256223. LINKS EXAMPLE a(3) = 5 because we obtain 5 following subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6. MATHEMATICA <<"DiscreteMath`Combinatorica`"; maxN=22; For[cnt=0; 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]], cnt++ ]]; Print[cnt]] PROG (Python) from math import gcd, lcm from itertools import combinations def A256220(n): m = lcm(*range(1, n+1)) fibset, mlist = set(), tuple(m//i for i in range(1, n+1)) a, b, c, k = 0, 1, 0, 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 s//gcd(s, m) in fibset: c += 1 if 2*l != n and (k-s)//gcd(k-s, m) in fibset: c += 1 return c+int(k//gcd(k, m) in fibset) # Chai Wah Wu, Feb 15 2022 CROSSREFS Cf. A000045, A075188, A010056, A256221, A256222, A256223. Sequence in context: A231716 A113488 A092917 * A163778 A328643 A160358 Adjacent sequences: A256217 A256218 A256219 * A256221 A256222 A256223 KEYWORD nonn,more AUTHOR Michel Lagneau, Mar 19 2015 EXTENSIONS a(23)-a(36) from Lars Blomberg, May 06 2015 STATUS approved

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Last modified January 31 03:33 EST 2023. Contains 359947 sequences. (Running on oeis4.)