A060764 -id:A060764 - cp's OEIS frontend

A060763 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n which are not also divisors of n.

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 3, 1, 1, 0, 2, 1, 3, 1, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 1, 1, 0, 2, 2, 3, 1, 1, 0, 3, 2, 3, 1, 1, 0, 1, 1, 4, 0, 3, 1, 1, 1, 3, 3, 1, 0, 1, 1, 3, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 0, 1, 2, 4, 0, 1, 1, 1, 1, 5

Offset: 1

Labos Elemer, Apr 24 2001

nonn

			For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; the differences {3,4,21} are not divisors, so a(70)=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060682, A060738, A060764.

f:= proc(n) local D,L;
  D:= numtheory:-divisors(n);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) minus D);
end proc:
map(f, [$1..200]); # Robert Israel, Jul 03 2017

a[n_] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]

a(n) = my(d=divisors(n)); #select(x->(setsearch(d, x)==0), vecsort(vector(#d-1, k, d[k+1] - d[k]),,8)); \\ Michel Marcus, Jul 04 2017

Edited by Dean Hickerson, Jan 22 2002

A060741 Number of divisors of 2n which are also differences between consecutive divisors of 2n (ordered by size).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 4, 5, 2, 5, 2, 4, 4, 4, 2, 5, 3, 4, 4, 4, 2, 7, 2, 6, 4, 4, 3, 6, 2, 4, 4, 5, 2, 5, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 7, 4, 4, 4, 4, 2, 9, 2, 4, 5, 7, 4, 5, 2, 6, 4, 6, 2, 8, 2, 4, 6, 6, 2, 6, 2, 6, 5, 4, 2, 8, 4, 4, 4, 6, 2, 9, 2, 6, 4, 4, 4, 7, 2, 6, 6, 6, 2, 6, 2, 6, 8

Offset: 1

Labos Elemer, Apr 23 2001

nonn

For odd numbers the intersection is empty.

			For n=35, 2n=70; divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; intersection={1,2,35}, so a(35)=3.

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

Cf. A000005, A060742, A060764.

a[n_ ] := Length[Intersection[Drop[d=Divisors[2n], 1]-Drop[d, -1], d]]
Table[Length[Intersection[Divisors[2n],Differences[Divisors[2n]]]],{n,110}] (* Harvey P. Dale, Nov 22 2015 *)

A060764(n) = { my(divs=divisors(n), diffs=vecsort(vector(#divs-1,i,divs[i+1]-divs[i]), ,8), c=#divs); for(i=1,#diffs,if(!(n%diffs[i]),c--)); (c); };
A060741(n) = (numdiv(2*n) - A060764(2*n)); \\ Antti Karttunen, Sep 21 2018

a(n) = A000005(2n) - A060764(2n).

Edited by Dean Hickerson, Jan 22 2002

A360118 Number of differences (not all necessarily distinct) between consecutive divisors of n which are not also divisors of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 3, 0, 1, 0, 1, 0, 3, 1, 1, 0, 2, 1, 3, 1, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 1, 5, 1, 1, 0, 2, 2, 3, 1, 1, 0, 3, 2, 3, 1, 1, 0, 1, 1, 5, 0, 3, 1, 1, 1, 3, 4, 1, 0, 1, 1, 5, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 0, 1, 2, 5, 0, 1, 1, 1, 1, 7

Offset: 1

Antti Karttunen, Feb 20 2023

nonn

			For n=70, its divisors are {1, 2, 5, 7, 10, 14, 35, 70} and their first differences are {1, 3, 2, 3, 4, 21, 35}, of which the differences {3, 3, 4, 21} are not divisors, so a(70) = 4. Note that in contrast to A060763, here the difference 3 is counted twice because there are two copies of it among the differences.

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

Cf. also A060763, A060764, A138652, A360119.

A360118(n) = { my(d=divisors(n), erot = vector(#d-1, k, d[k+1] - d[k])); sum(i=1,#erot,!!(n%erot[i])); };

For all n >= 1, A060763(n) <= a(n) < A060764(n).

a(n) = A060764(n) - A360119(n).

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.

Original entry on oeis.org

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).

A060700 "Anomalous" numbers k such that for even numbers 2k, gcd(2k, lcm(dd(2k)))=2k and not k, where dd(2k) is the first difference set of divisors of 2k.

Original entry on oeis.org

15, 30, 35, 45, 63, 70, 75, 77, 91, 99, 105, 117, 126, 135, 140, 143, 150, 153, 154, 165, 175, 182, 187, 189, 195, 198, 209, 221, 225, 231, 234, 245, 247, 252, 255, 273, 280, 285, 286, 297, 299, 306, 308, 315, 323, 325, 330, 345, 350, 351, 357, 364, 374, 375

Offset: 1

Labos Elemer, Apr 25 2001

nonn

			63 is here because for 126 = 2*63, lcm(dd(126)) = lcm(1, 1, 3, 1, 2, 5, 4, 3, 21, 21, 63) = 1260, so gcd(126, lcm(dd(126))) = gcd(126, 1260) = 126.

Cf. A060680, A060681, A060682, A060683, A060684, A060685, A060741, A060742, A060763, A060764, A060765, A060766, A000005.

f(n) = {my(d = divisors(n), dd = vector(#d-1, k, d[k+1] - d[k])); gcd(n, lcm(dd));}
isok(n) = (f(2*n) == 2*n); \\ Michel Marcus, Mar 29 2018

cp's OEIS Frontend

A060763 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n which are not also divisors of n.

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A060741 Number of divisors of 2n which are also differences between consecutive divisors of 2n (ordered by size).

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A360118 Number of differences (not all necessarily distinct) between consecutive divisors of n which are not also divisors of n.

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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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A060700 "Anomalous" numbers k such that for even numbers 2k, gcd(2k, lcm(dd(2k)))=2k and not k, where dd(2k) is the first difference set of divisors of 2k.

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PARI