A139711 - OEIS

%I #27 Mar 17 2023 05:27:40

%S 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,

%T 39,41,43,45,47,48,49,51,53,55,57,59,60,61,63,64,65,67,69,71,73,75,77,

%U 79,80,81,83,85,87,89,91,93,95,96,97,99,100,101,103,105,107,109,111,112

%N 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.

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

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

%H Harvey P. Dale, <a href="/A139711/b139711.txt">Table of n, a(n) for n = 1..1000</a>

%F {k: A000035(A033676(k) + A033677(k)) = 0}. - _R. J. Mathar_, May 11 2008

%e 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.

%p 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

%t 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 *)

%t Select[Range[112],IntegerQ[Median[Divisors[#]]] &] (* _Stefano Spezia_, Mar 14 2023 *)

%o (PARI) A063655(n) = {local(d); d=divisors(n); d[(length(d)+1)\2] + d[length(d)\2+1]};

%o for(n=1, 120, if(A063655(n)%2==0, print1(n, ", ")) ) \\ _G. C. Greubel_, May 31 2019

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

%K nonn

%O 1,2

%A _Leroy Quet_, Apr 30 2008

%E More terms from _R. J. Mathar_, May 11 2008