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

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, 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, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Author

Michel Marcus, Oct 11 2012

Keywords

• nonn

Comments

a(n) <= A111233(n).

a(n) <= floor(Sum_{k=1..n} 1/k) = A055980(n). - Joerg Arndt, Oct 13 2012

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

a(k) = n for all k such that A101877(n) <= k < A101877(n+1). - Jon E. Schoenfield, May 12 2017

Examples

			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.
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.
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
		

References

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

Links

• Manfred Scheucher, Sage Script
• H. Yokota, On number of integers representable as sums of unit fractions, Canad. Math. Bull. Vol. 33 (2), 1990.
• H. Yokota, On Number of Integers Representable as a Sum of Unit Fractions, II, Journal of Number Theory 67, 162-169, 1997.

Crossrefs

Cf. A101877, A111233.

Programs

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))); }

Extensions

a(25)-a(46) from Manfred Scheucher, Aug 17 2015

a(47)-a(87) from Jon E. Schoenfield, Apr 30 2018