cp's OEIS Frontend

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.

%I A139710 #24 Mar 17 2023 05:27:06
%S A139710 2,6,10,12,14,18,20,22,26,28,30,34,38,40,42,44,46,50,52,54,56,58,62,
%T A139710 66,68,70,72,74,76,78,82,84,86,88,90,92,94,98,102,104,106,108,110,114,
%U A139710 116,118,122,124,126,130,132,134,136,138,142,146,148,150,152,154,156,158
%N 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.
%C A139710 All terms of this sequence are even.
%C A139710 A139711 contains all positive integers not in this sequence and vice versa.
%H A139710 G. C. Greubel, <a href="/A139710/b139710.txt">Table of n, a(n) for n = 1..10000</a>
%F A139710 {k: A000035(A033676(k) + A033677(k)) = 1}. - _R. J. Mathar_, May 11 2008
%e A139710 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.
%p A139710 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
%t A139710 centralDivisors:=#[[({Floor[#],Ceiling[#]}&[(1+#)/2&[Length[#]]])]]&[Divisors[#]]&;
%t A139710 Select[Range[500],OddQ[Total[#]]&[centralDivisors[#]]&](* _Peter J. C. Moses_, May 31 2019 *)
%t A139710 Select[Range[158],!IntegerQ[Median[Divisors[#]]] &] (* _Stefano Spezia_, Mar 14 2023 *)
%o A139710 (PARI) b(n) = {local(d); d=divisors(n); d[(length(d)+1)\2] + d[length(d)\2+1]};
%o A139710 for(n=1, 180, if(b(n)%2==1, print1(n, ", ")) ) \\ _G. C. Greubel_, May 31 2019
%Y A139710 Cf. A000035, A033676, A033677, A063655, A139711.
%K A139710 nonn
%O A139710 1,1
%A A139710 _Leroy Quet_, Apr 30 2008
%E A139710 More terms from _R. J. Mathar_, May 11 2008