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

1,2

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.

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A039065 A247786 A330565 * A354994 A214585 A196036

Adjacent sequences: A139708 A139709 A139710 * A139712 A139713 A139714

Keyword

nonn

Author

Leroy Quet, Apr 30 2008

Extensions

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

Status

approved