A050205
Triangle read by rows: number of terms in unit fraction representation of k/n using the greedy algorithm, 1<=k<=n-1.
Original entry on oeis.org
1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 3, 3, 1, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 4, 4, 4, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3, 3, 4, 1, 2, 1, 2, 1, 2, 3, 2, 2, 2, 3
Offset: 2
2/3 = 1/2 + 1/6. So T(3,2) = 2.
n\k | 1 2 3 4 5 6 7 8
----*------------------------
2 | 1;
3 | 1, 2;
4 | 1, 1, 2;
5 | 1, 2, 2, 3;
6 | 1, 1, 1, 2, 2;
7 | 1, 2, 3, 2, 3, 3;
8 | 1, 1, 2, 1, 2, 2, 3;
9 | 1, 2, 1, 2, 2, 2, 3, 3;
A050206
Triangle read by rows: smallest denominator of the expansion of k/n using the greedy algorithm, 1<=k<=n-1.
Original entry on oeis.org
2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 6, 3, 2, 2, 2, 7, 4, 3, 2, 2, 2, 8, 4, 3, 2, 2, 2, 2, 9, 5, 3, 3, 2, 2, 2, 2, 10, 5, 4, 3, 2, 2, 2, 2, 2, 11, 6, 4, 3, 3, 2, 2, 2, 2, 2, 12, 6, 4, 3, 3, 2, 2, 2, 2, 2, 2, 13, 7, 5, 4, 3, 3, 2, 2, 2, 2, 2, 2, 14, 7, 5, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 15, 8, 5, 4, 3, 3, 3, 2
Offset: 2
n\k | 1 2 3 4 5 6 7 8
----*------------------------
2 | 2;
3 | 3, 2;
4 | 4, 2, 2;
5 | 5, 3, 2, 2;
6 | 6, 3, 2, 2, 2;
7 | 7, 4, 3, 2, 2, 2;
8 | 8, 4, 3, 2, 2, 2, 2;
9 | 9, 5, 3, 3, 2, 2, 2, 2;
A260618
Irregular triangle read by rows: denominators of the expansion of k/n using the greedy algorithm, 1<=k<=n.
Original entry on oeis.org
1, 2, 1, 3, 2, 6, 1, 4, 2, 2, 4, 1, 5, 3, 15, 2, 10, 2, 4, 20, 1, 6, 3, 2, 2, 6, 2, 3, 1, 7, 4, 28, 3, 11, 231, 2, 14, 2, 5, 70, 2, 3, 42, 1, 8, 4, 3, 24, 2, 2, 8, 2, 4, 2, 3, 24, 1, 9, 5, 45, 3, 3, 9, 2, 18, 2, 6, 2, 4, 36, 2, 3, 18, 1, 10, 5, 4, 20, 3, 15, 2, 2, 10, 2, 5, 2, 4, 20, 2, 3, 15, 1, 11, 6, 66, 4, 44, 3, 33, 3, 9, 99, 2, 22, 2, 8, 88, 2, 5, 37, 4070, 2, 4, 15, 660, 2, 3, 14, 231, 1
Offset: 1
Triangle begins ({} included for fraction separation):
{1};
{2}, {1};
{3}, {2, 6}, {1};
{4}, {2}, {2, 4}, {1};
{5}, {3, 15}, {2, 10}, {2, 4, 20}, {1};
{6}, {3}, {2}, {2, 6}, {2, 3}, {1};
{7}, {4, 28}, {3, 11, 231}, {2, 14}, {2, 5, 70}, {2, 3, 42}, {1};
{8}, {4}, {3, 24}, {2}, {2, 8}, {2, 4}, {2, 3, 24}, {1};
{9}, {5, 45}, {3}, {3, 9}, {2, 18}, {2, 6}, {2, 4, 36}, {2, 3, 18}, {1};
{10}, {5}, {4, 20}, {3, 15}, {2}, {2, 10}, {2, 5}, {2, 4, 20}, {2, 3, 15}, {1};
{11}, {6, 66}, {4, 44}, {3, 33}, {3, 9, 99}, {2, 22}, {2, 8, 88}, {2, 5, 37, 4070}, {2, 4, 15, 660}, {2, 3, 14, 231}, {1};
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rep(f)={L=List(); while(f<>0, my(t=ceil(1/f)); listput(L,t); f-=1/t); Vec(L)}
row(n)={concat(apply(k->rep(k/n), [1..n]))}
for(n=1, 11, print(row(n))) \\ Andrew Howroyd, Feb 26 2018
A100140
Largest denominator of greedy Egyptian fraction sum for M/N.
Original entry on oeis.org
2, 6, 4, 20, 6, 231, 24, 45, 20, 4070, 12, 2145, 231, 120, 240, 3039345, 45, 2359420, 180, 1428, 4070, 1019084, 120, 53307975, 2145, 1350, 1428, 1003066152, 120, 1127619917796295, 16800, 26796, 3039345, 1104740, 72, 884004, 2359420, 1288092
Offset: 2
Consider a(5). There are 4 fractions with 5 in the denominator: 1/5=1/5, 2/5=1/3+1/15, 3/5=1/2+1/10 and 4/5=1/2+1/4+1/20. Of these, the largest denominator is 20, so a(5)=20.
- R. K. Guy, "Egyptian Fractions." section D11 in "Unsolved Problems in Number Theory", 2nd ed. New York: Springer-Verlag, pp. 158-166, 1994.
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/* MACSYMA or maxima */ egypt(x) := block([i,n,d,p,e, on, od], ( n : num(x), d : n/x, on : n, od : d, p : 0, e : [], for i:1 while x>0 do ( n : num(x), d : n/x, p : fix((d+n-1)/n), x : x - 1/p, e : append(e, [p]) ), return(p) ) ); for b:2 step 1 thru 100 do ( max:2, for a:2 step 1 thru b-1 do ( if gcd(a,b)=1 then ( m : egypt(a/b), if m>max then max : m ) ), print("a[", b, "]=", max) ), t$
A091834
Triangle read by rows: T(n,k) = maximum denominator in the Egyptian fraction representation that minimizes the maximum denominator of the k-th entry of row n of the triangle of numbers 1/2; 1/3, 2/3; 1/4, 2/4, 3/4; ...
Original entry on oeis.org
2, 3, 6, 4, 2, 4, 5, 15, 10, 10, 6, 3, 2, 6, 3, 7, 21, 21, 14, 14, 21, 8, 4, 8, 2, 8, 4, 8, 9, 18, 3, 9, 18, 6, 9, 18, 10, 5, 10, 15, 2, 10, 5, 10, 15, 11, 44, 44, 33, 33, 22, 22, 44, 44, 33
Offset: 0
The 47th position, corresponding to the fraction 2/11, has value 44 because 2/11=1/12+1/22+1/33+1/44 and there is no way of writing 2/11 as a sum of distinct unit fractions with all denominators less than 44.
A362289
a(n) is the largest denominator when the greedy algorithm for Egyptian fractions is applied to 1/n + 1/(n+1).
Original entry on oeis.org
2, 3, 12, 180, 30, 1428, 56, 2520, 90, 2310, 132, 100292556, 182, 9240, 240, 119952, 306, 614444040, 380, 23100, 462, 42190274940, 552, 77390453400, 650, 201474, 756, 23370247110, 870, 200880, 992, 14523137084239067683872, 1122, 2206260, 1260, 104845560637757648698080
Offset: 1
For n=16, 1/16 + 1/17 = 33/272 which written in Egyptian fractions is 1/9 + 1/98 + 1/119952 and the largest denominator is 119952.
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egyptFraction[f_] := Ceiling[1/Most[NestWhileList[# - 1/Ceiling[1/#] &, f, # != 0 &]]]; a[n_] := egyptFraction[1/n + 1/(n + 1)][[-1]]; Array[a, 40] (* Amiram Eldar, Apr 14 2023 *)
Showing 1-6 of 6 results.
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