A281527 -id:A281527 - cp's OEIS frontend

A281530 Triangle read by rows: T(n,k) = number of terms for the shortest Egyptian fraction representation of k/n, 1 <= k < n.

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, 3, 3, 3

Offset: 2

Arkadiusz Wesolowski, Jan 23 2017

Not same as A050205. Example: the fraction 9/20 requires three terms in its greedy expansion, but 9/20 = 1/4 + 1/5, so T(20,9) = 2.

			The triangle T(n,k) begins:
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

Index entries for sequences related to Egyptian fractions

Cf. A281527.

A281529 a(n) = least denominator Y of any proper fraction X/Y which needs n or more terms to be written as a signed sum of distinct unit fractions.

Original entry on oeis.org

2, 3, 7, 23, 163

Offset: 1

Arkadiusz Wesolowski, Jan 23 2017

W. Sierpiński, O rozkładach liczb wymiernych na ułamki proste, PWN, Warsaw, Poland, 1957, pp. 96-100.

Cf. A281527. See A281532 for numerators.

A281528 a(n) = least numerator k such that the proper fraction k/n needs three or more terms as a signed sum of distinct unit fraction, or 0 if no such numerator exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 0, 7, 0, 7, 0, 5, 10, 11, 11, 5, 13, 7, 13, 13, 14, 5, 17, 7, 10, 11, 17, 8, 19, 7, 13, 13, 10, 11, 23, 5, 11, 11, 17, 9, 25, 5, 17, 13, 10, 5, 29, 9, 14, 11, 19, 5, 22, 13, 17, 13, 11, 7, 37, 7, 13, 13, 19, 17, 26, 5, 20, 15, 22, 11, 29, 5, 10

Offset: 2

Arkadiusz Wesolowski, Jan 23 2017

nonn

Index entries for sequences related to Egyptian fractions

Cf. A281527.

lst:=[]; for n in [2..74] do for k in [1..n-1] do f:=k/n; x:=1; v:=0; if Numerator(f) eq 1 then v:=1; else while f lt 2/x do if Numerator(Abs(f-1/x)) eq 1 then v:=1; break; end if; x+:=1; end while; end if; if v eq 0 then Append(~lst, k); break; end if; if k eq n-1 then Append(~lst, 0); end if; end for; end for; lst;

cp's OEIS Frontend

A281530 Triangle read by rows: T(n,k) = number of terms for the shortest Egyptian fraction representation of k/n, 1 <= k < n.

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A281529 a(n) = least denominator Y of any proper fraction X/Y which needs n or more terms to be written as a signed sum of distinct unit fractions.

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A281528 a(n) = least numerator k such that the proper fraction k/n needs three or more terms as a signed sum of distinct unit fraction, or 0 if no such numerator exists.

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