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 A217693 Numbers of distinct integers obtained from summing up subsets of {1, 1/2, 1/3, ..., 1/n}. 1

%I

%S 1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,

%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,

%U 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4

%N Numbers of distinct integers obtained from summing up subsets of {1, 1/2, 1/3, ..., 1/n}.

%C a(n) <= A111233(n).

%C a(n) <= floor(Sum_{k=1..n} 1/k) = A055980(n). - _Joerg Arndt_, Oct 13 2012

%C a(n) <= 4 for n <= 94, a(n) <= 5 for n <= 257, a(n) <= 6 for n <= 689. That is because if there is a term 1/a with p dividing a for a prime p, then there must be another term 1/b with p dividing b. Hence, not all terms from 1/1 to 1/n can be summed up. Cf. the "filter" function in my Sage script. - _Manfred Scheucher_, Aug 17 2015

%C a(k) = n for all k such that A101877(n) <= k < A101877(n+1). - _Jon E. Schoenfield_, May 12 2017

%D P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory, Université de Genève, 1980.

%H Manfred Scheucher, <a href="/A217693/a217693.sage.txt">Sage Script</a>

%H H. Yokota, <a href="http://dx.doi.org/10.4153/CMB-1990-037-0">On number of integers representable as sums of unit fractions</a>, Canad. Math. Bull. Vol. 33 (2), 1990.

%H H. Yokota, <a href="http://dx.doi.org/10.1006/jnth.1997.2187">On Number of Integers Representable as a Sum of Unit Fractions, II</a>, Journal of Number Theory 67, 162-169, 1997.

%e 1, 1/2 + 1/3 + 1/6 = 1 and 1 + 1/2 + 1/3 + 1/6 = 2 are integers, but only 2 of them are distinct, so a(6)=2.

%e a(24)=3 because 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/15 + 1/18 + 1/20 + 1/24 = 3 and Sum_{k=1..n} 1/k < 4 for all n <= 30.

%e a(65)=4 because the sum of the reciprocals of the integers in { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 27, 28, 30, 33, 35, 36, 40, 42, 45, 48, 52, 54, 56, 60, 63, 65 } is 4 and Sum_{k=1..n} 1/k < 5 for all n <= 82. - _Jon E. Schoenfield_, Apr 30 2018

%o (PARI) ufr(n) = {tab = []; for (i=1, 2^n - 1, vb = binary(i); while(length(vb) < n, vb = concat(0, vb););; val = sum(j=1, length(vb), vb[j]/j); if (denominator(val) == 1, tab = concat(tab, val); ); ); return (length(Set(tab))); }

%Y Cf. A101877, A111233.

%K nonn

%O 1,6

%A _Michel Marcus_, Oct 11 2012

%E a(25)-a(46) from _Manfred Scheucher_, Aug 17 2015

%E a(47)-a(87) from _Jon E. Schoenfield_, Apr 30 2018

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Last modified January 29 01:42 EST 2020. Contains 331328 sequences. (Running on oeis4.)