

A138652


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


2



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
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OFFSET

1,2


COMMENTS

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


LINKS



FORMULA



EXAMPLE

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.
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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(i1, 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 *)


PROG

(PARI) A138652(n) = { n = 2*n; my(d=divisors(n), erot = vector(#d1, k, d[k+1]  d[k])); sum(i=1, #erot, !(n%erot[i])); }; \\ Antti Karttunen, Feb 20 2023


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



