A139711 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 even.

1, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 36, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 80, 81, 83, 85, 87, 89, 91, 93, 95, 96, 97, 99, 100, 101, 103, 105, 107, 109, 111, 112

Offset: 1

Author

Leroy Quet, Apr 30 2008

Keywords

• nonn

Comments

All odd positive integers and all perfect squares are included in this sequence.

A139710 contains all positive integers not in this sequence and vice versa.

Examples

			The divisors of 24 are 1,2,3,4,6,8,12,24. The middle 2 divisors are 4 and 6. The sum of these is 10, which is even. So 24 is included in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000035, A033676, A033677, A063655, A139710.

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: isA139711 := proc(n) RETURN ( ( A033676(n)+A033677(n) ) mod 2 = 0 ) ; end: for n from 1 to 300 do if isA139711(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 11 2008

Mathematica

evdQ[n_]:=Module[{divs=Divisors[n],sr=Sqrt[n]},EvenQ[Max[Select[divs, #<=sr&]]+Min[Select[divs,#>=sr&]]]]; Select[Range[120],evdQ] (* Harvey P. Dale, Mar 05 2012 *)
Select[Range[112],IntegerQ[Median[Divisors[#]]] &] (* Stefano Spezia, Mar 14 2023 *)

PARI

A063655(n) = {local(d); d=divisors(n); d[(length(d)+1)\2] + d[length(d)\2+1]};
for(n=1, 120, if(A063655(n)%2==0, print1(n, ", ")) ) \\ G. C. Greubel, May 31 2019

Formula

{k: A000035(A033676(k) + A033677(k)) = 0}. - R. J. Mathar, May 11 2008

Extensions

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