A104514 a(n) = least number k > 1 of consecutive integers which sum to 2*n; or a(n) = 0 if n is a power of 2.

0, 0, 3, 0, 4, 3, 4, 0, 3, 5, 4, 3, 4, 7, 3, 0, 4, 3, 4, 5, 3, 8, 4, 3, 4, 8, 3, 7, 4, 3, 4, 0, 3, 8, 4, 3, 4, 8, 3, 5, 4, 3, 4, 11, 3, 8, 4, 3, 4, 5, 3, 13, 4, 3, 4, 7, 3, 8, 4, 3, 4, 8, 3, 0, 4, 3, 4, 16, 3, 5, 4, 3, 4, 8, 3, 16, 4, 3, 4, 5, 3, 8, 4, 3, 4, 8, 3, 11, 4, 3, 4, 16, 3, 8, 4, 3, 4, 7, 3, 5, 4, 3, 4

Offset: 1

Author

Alfred S. Posamentier (asp2(AT)juno.com) and Robert G. Wilson v, Feb 23 2005

Keywords

• nonn

Comments

a(2^k) = 0 and a(3*n) = 3.

Least proper divisor d of 4*n (if any) such that d or 4*n/d is odd. - Robert Israel, May 06 2015

Examples

			a(9) = 3 because 3+4+5+6 = 5+6+7 = 2*9 = 18.
		

References

• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 67.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A104512, A104513, A104515, A104516, A163169.

Programs

Maple

a:= proc(n) local divs,r;
   divs:= select(t -> t::odd or (4*n/t)::odd, numtheory:-divisors(4*n) minus {1,4*n});
   if nops(divs)=0 then 0 else min(divs) fi
end proc:
seq(a(n), n=1..200); # Robert Israel, May 06 2015

Mathematica

f[n_] := Block[{r = Ceiling[n/2]}, If[IntegerQ[Log[2, n]], 0, m = Range[r]; lst = Flatten[Table[m[[k]], {i, r}, {j, i + 1, r}, {k, i, j}], 1]; Min[Length /@ lst[[Flatten[Position[Plus @@@ lst, n]]]]]]]; Table[f[2n], {n, 103}]

Formula

a(n) = A163169(2*n). Robert Israel, May 06 2015