A098853 -id:A098853 - cp's OEIS frontend

A260618 Irregular triangle read by rows: denominators of the expansion of k/n using the greedy algorithm, 1<=k<=n.

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

Matthew Campbell, Sep 17 2015

			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};

Seiichi Manyama, Rows n=1..81 of triangle, flattened (Rows 1..29 from Andrew Howroyd)
Wikipedia, Greedy algorithm for Egyptian fractions

Cf. A050205, A050206, A050210, A098853, A100140.

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

Robert Munafo, Nov 06 2004

nonn

Each term gives the largest of the N-1 terms in A050210 corresponding to the fractions with denominator N.

			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.

Seiichi Manyama, Table of n, a(n) for n = 2..198
Robert Munafo, Largest Denominator of Greedy Egyptian Fraction Sum for M/N
Eric Weisstein's World of Mathematics, Egyptian Fractions.

Cf. A050210, A098853, A260618.

/* 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$

a(6) corrected by Seiichi Manyama, Sep 18 2022

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A260618 Irregular triangle read by rows: denominators of the expansion of k/n using the greedy algorithm, 1<=k<=n.

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A100140 Largest denominator of greedy Egyptian fraction sum for M/N.

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