A360118 -id:A360118 - cp's OEIS frontend

A360119 Number of divisors of n which are not also differences between consecutive divisors, minus the number of differences between consecutive divisors of n which are not also divisors of n. Here the differences are counted with repetition if they occur more than once.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 6, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 3, 1, 1, 1

Offset: 1

Antti Karttunen, Feb 20 2023

nonn

Because the algorithm for computing this sequence (see the PARI program) starts with s set to the number of divisors, and s is decremented at most once on each iteration in the loop over the first differences of the divisors, and because there is one less difference than there are divisors, it implies that a(n) >= 1 for all n.

Note that if a(n) = 1, then A088722(n) = 0, but not vice versa, i.e., the positions of 1's in this sequence is just a subsequence of A088725. See A360129 for the exceptions.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A060764, A088722, A088725, A360118, A360129.

A360119(n) = { my(d=divisors(n), erot=vecsort(vector(#d-1, k, d[k+1] - d[k])), s=#d); for(i=1,#erot,if(n%erot[i], s--, if(1==i || erot[i]!=erot[i-1], s--))); (s); };

a(n) = A060764(n) - A360118(n).

a(n) <= A000005(n).

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

Original entry on oeis.org

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

Leroy Quet, May 15 2008

nonn

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

			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)

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

Cf. A060741, A060763, A066660, A360118.

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

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 *)

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

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

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

Definition edited and clarified by Antti Karttunen, Feb 20 2023

cp's OEIS Frontend

A360119 Number of divisors of n which are not also differences between consecutive divisors, minus the number of differences between consecutive divisors of n which are not also divisors of n. Here the differences are counted with repetition if they occur more than once.

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