A138652 Number of differences (not all necessarily distinct) between consecutive divisors of 2n which are also divisors of 2n.

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, 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, 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

Offset: 1

Author

Leroy Quet, May 15 2008

Keywords

• nonn

Comments

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

Examples

			From _Antti Karttunen_, Feb 20 2023: (Start)
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.
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.
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.
(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A060741, A060763, A066660, A360118.

Programs

Maple

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

Mathematica

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 *)

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

Formula

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

Extensions

More terms from Stefan Steinerberger and R. J. Mathar, May 18 2008

Definition edited and clarified by Antti Karttunen, Feb 20 2023