A139710 Numbers k such that the sum of (the largest divisor of k that is <= sqrt(k)) and (the smallest divisor of k that is >= sqrt(k)) is odd.

2, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 38, 40, 42, 44, 46, 50, 52, 54, 56, 58, 62, 66, 68, 70, 72, 74, 76, 78, 82, 84, 86, 88, 90, 92, 94, 98, 102, 104, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 136, 138, 142, 146, 148, 150, 152, 154, 156, 158

Offset: 1

Author

Leroy Quet, Apr 30 2008

Keywords

• nonn

Comments

All terms of this sequence are even.

A139711 contains all positive integers not in this sequence and vice versa.

Examples

			The divisors of 12 are 1,2,3,4,6,12. The middle 2 divisors are 3 and 4. The sum of these is 7, which is odd. So 12 is included in the sequence.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000035, A033676, A033677, A063655, A139711.

Programs

Maple

A033676 := proc(n) local d ; for d from floor(sqrt(n)) to 1 by -1 do if n mod d = 0 then RETURN(d) ; fi ; od: end: A033677 := proc(n) n/A033676(n) ; end: isA139710 := proc(n) RETURN ( ( A033676(n)+A033677(n) ) mod 2 = 1 ) ; end: for n from 1 to 300 do if isA139710(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 11 2008

Mathematica

centralDivisors:=#[[({Floor[#],Ceiling[#]}&[(1+#)/2&[Length[#]]])]]&[Divisors[#]]&;
Select[Range[500],OddQ[Total[#]]&[centralDivisors[#]]&](* Peter J. C. Moses, May 31 2019 *)
Select[Range[158],!IntegerQ[Median[Divisors[#]]] &] (* Stefano Spezia, Mar 14 2023 *)

PARI

b(n) = {local(d); d=divisors(n); d[(length(d)+1)\2] + d[length(d)\2+1]};
for(n=1, 180, if(b(n)%2==1, print1(n, ", ")) ) \\ G. C. Greubel, May 31 2019

Formula

{k: A000035(A033676(k) + A033677(k)) = 1}. - R. J. Mathar, May 11 2008

Extensions

More terms from R. J. Mathar, May 11 2008