A281531 a(n) is the least numerator k such that the proper fraction k/n needs three or more terms as an Egyptian fraction, or 0 if no such numerator exists.

0, 0, 0, 4, 0, 3, 7, 7, 8, 5, 11, 3, 6, 7, 7, 4, 13, 3, 13, 9, 5, 5, 17, 4, 6, 8, 12, 4, 14, 3, 7, 5, 8, 11, 17, 3, 6, 9, 17, 4, 18, 3, 7, 11, 7, 5, 21, 3, 8, 7, 11, 4, 13, 9, 13, 7, 7, 7, 28, 3, 5, 13, 7, 4, 10, 3, 11, 11, 13, 5, 23, 3, 6, 11, 9, 5, 11, 3, 19

Offset: 2

Author

Arkadiusz Wesolowski, Jan 23 2017

Keywords

• nonn

Comments

If n > 3 is prime, a(n) = A007978(n+1). - Robert Israel, Dec 26 2019

Links

• Robert Israel, Table of n, a(n) for n = 2..10000
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A007978, A281530.

Programs

Magma

lst:=[]; for n in [2..80] 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(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;

Maple

f:= proc(n) option remember; local k,T;
  T:= numtheory:-divisors(n^2);
  for k from 2 to n-1 do
    g:= igcd(k,n);
    if g > 1 then
       r:= procname(n/g);
       if k = r*g then return k fi;
    else
       if not member(-n mod k,  T mod k) then return k fi
    fi
  od;
0
end proc;
map(f, [$2..100]); # Robert Israel, Dec 25 2019

Mathematica

a[n_] := a[n] = Module[{k, T}, T = Divisors[n^2]; For[k = 2, k <= n - 1, k++, g = GCD[k, n]; If[g > 1, r = a[n/g]; If[k == r g, Return [k]], If[FreeQ[Mod[T, k], Mod[-n, k]], Return [k]]]]; 0];
a /@ Range[2, 100] (* Jean-François Alcover, Oct 06 2020, after Robert Israel *)