

A217693


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


1



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
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OFFSET

1,6


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


REFERENCES

P. Erdos and R. L. Graham, Old and new problems and results in combinatorial number theory, UniversitÃ© de GenÃ¨ve, 1980.


LINKS

Table of n, a(n) for n=1..87.
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, 162169, 1997.


EXAMPLE

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


PROG

(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))); }


CROSSREFS

Cf. A101877, A111233.
Sequence in context: A194338 A176170 A062153 * A204560 A135661 A082998
Adjacent sequences: A217690 A217691 A217692 * A217694 A217695 A217696


KEYWORD

nonn


AUTHOR

Michel Marcus, Oct 11 2012


EXTENSIONS

a(25)a(46) from Manfred Scheucher, Aug 17 2015
a(47)a(87) from Jon E. Schoenfield, Apr 30 2018


STATUS

approved



