%I #20 Feb 20 2023 21:49:44

%S 1,2,3,3,2,5,2,4,5,5,2,7,2,4,6,5,2,8,2,6,7,4,2,9,3,4,7,5,2,11,2,6,6,4,

%T 3,11,2,4,6,7,2,10,2,6,10,4,2,11,3,8,6,6,2,11,5,6,6,4,2,15,2,4,9,7,4,

%U 9,2,6,6,8,2,14,2,4,9,6,2,11,2,8,9,4,2,15,4,4,6,6,2,17,3,6,6,4,4,13,2,6,9

%N Number of differences (not all necessarily distinct) between consecutive divisors of 2n which are also divisors of 2n.

%C For n = any odd positive integer, there are no differences (between consecutive divisors of n) that divide n.

%H Antti Karttunen, <a href="/A138652/b138652.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) + A360118(2n) = A000005(2n)-1, i.e., a(n) = A066660(n) - A360118(2*n). - Reference to a wrong A-number replaced with A360118 by _Antti Karttunen_, Feb 20 2023

%e From _Antti Karttunen_, Feb 20 2023: (Start)

%e Divisors of 2*12 = 24 are: [1, 2, 3, 4, 6, 8, 12, 24]. Their first differences are: [1, 1, 1, 2, 2, 4, 12], all 7 which are divisors of 24, thus a(12) = 7.

%e Divisors of 2*35 = 70 are: [1, 2, 5, 7, 10, 14, 35, 70]. Their first differences are: 1, 3, 2, 3, 4, 21, 35, of which 1, 2 and 35 are divisors of 70, thus a(35) = 3.

%e Divisors of 2*65 = 130 are: [1, 2, 5, 10, 13, 26, 65, 130]. Their first differences are: 1, 3, 5, 3, 13, 39, 65, of which 1, 5, 13 and 65 are divisors of 130, thus a(65) = 4.

%e (End)

%p A138652 := proc(n) local a,dvs,i ; a := 0 ; dvs := sort(convert(numtheory[divisors](2*n),list)) ; for i from 2 to nops(dvs) do if (2*n) mod ( op(i,dvs)-op(i-1,dvs) ) = 0 then a := a+1 ; fi ; od: a ; end: seq(A138652(n),n=1..120) ; # _R. J. Mathar_, May 20 2008

%t a = {}; For[n = 2, n < 200, n = n + 2, b = Table[Divisors[n][[i + 1]] - Divisors[n][[i]], {i, 1, Length[Divisors[n]] - 1}]; AppendTo[a, Length[Select[b, Mod[n, # ] == 0 &]]]]; a (* _Stefan Steinerberger_, May 18 2008 *)

%o (PARI) A138652(n) = { n = 2*n; my(d=divisors(n), erot = vector(#d-1, k, d[k+1] - d[k])); sum(i=1,#erot,!(n%erot[i])); }; \\ _Antti Karttunen_, Feb 20 2023

%Y Cf. A060741, A060763, A066660, A360118.

%K nonn

%O 1,2

%A _Leroy Quet_, May 15 2008

%E More terms from _Stefan Steinerberger_ and _R. J. Mathar_, May 18 2008

%E Definition edited and clarified by _Antti Karttunen_, Feb 20 2023