A060763 -id:A060763 - cp's OEIS frontend

A060764 Number of divisors of n which are not also differences between consecutive divisors (ordered by increasing magnitude) of n.

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

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 divisors {5,7,10,14,70} are not differences, so a(70)=5.

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

Cf. A000005, A060741, A060763.

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

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); }; \\ Antti Karttunen, Sep 21 2018

from itertools import pairwise
from sympy import divisors
def A060764(n):
    e = map(lambda x:x[1]-x[0],pairwise(d:=divisors(n)))
    return len(set(d)-set(e)) # Chai Wah Wu, Feb 21 2023

Edited by Dean Hickerson, Jan 22 2002

A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 42, 48, 54, 60, 64, 72, 96, 100, 108, 120, 128, 144, 156, 162, 168, 180, 192, 216, 240, 256, 272, 288, 294, 300, 324, 342, 360, 384, 432, 480, 486, 500, 504, 512, 576, 600, 648, 720, 768, 840, 900, 960, 972, 1008, 1024

Offset: 1

Labos Elemer, Apr 24 2001

nonn

Equivalently, A060763(n)=0.
Powers of 2 and factorials up to 7! are here.
For each k=1..A000005(a(n))-1 exists k' < A000005(a(n)) such that A193829(a(n),k) = A027750(a(n),k'). - Reinhard Zumkeller, Jun 25 2015
From Robert Israel, Jul 03 2017: (Start)
Also includes 3*2^k and 2*3^k for all k>= 1.
All terms except 1 are even. (End)
Conjecture: a(n) has the property that for each prime divisor p, p-1|a(n)/p. If this conjecture is true then terms can be searched by distinct prime divisors. - David A. Corneth, Jul 06 2017
The divisors of a(n) form a Brauer chain. See A079301 for the definition of a Brauer chain. - Zizheng Fang, Jan 30 2020

			For n = 12, divisors={1, 2, 3, 4, 6, 12}; differences={1, 1, 1, 2, 6}; every difference is a divisor, so 12 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..750

Cf. A060683, A060763, A193829, A027750.

import Data.List (sort, nub); import Data.List.Ordered (subset)
a060765 n = a060765_list !! (n-1)
a060765_list = filter
(\x -> sort (nub $ a193829_row x) `subset` a027750_row' x) [1..]
-- Reinhard Zumkeller, Jun 25 2015

[k:k in [1..1025]| forall{i:i in [2..#Divisors(k)]|k mod (d[i]-d[i-1]) eq 0 where d is Divisors(k)}]; // Marius A. Burtea, Jan 30 2020

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:
select(f=0, [$1..1000]); # Robert Israel, Jul 03 2017

test[n_ ] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]==0; Select[Range[1, 1024], test]
(* Second program: *)
Select[Range[2^10], Function[n, AllTrue[Differences@ Divisors@ n, Divisible[n, #] &]]] (* Michael De Vlieger, Jul 12 2017 *)

isok(n)=my(d=divisors(n), v=vecsort(vector(#d-1, k, d[k+1]-d[k]),,8)); #select(x->setsearch(d, x), v) == #v; \\ Michel Marcus, Jul 06 2017

is(n)=my(t); fordiv(n,d, if(n%(d-t), return(0)); t=d); 1 \\ Charles R Greathouse IV, Jul 12 2017

Edited by Dean Hickerson, Jan 22 2002

A060738 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n! which are not also divisors of n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 4, 5, 12, 51, 92, 225, 340, 573, 1325, 2334, 6024, 8724, 13130, 21601, 46169, 67213, 106427, 178014, 242104, 338499, 727248, 988029, 1924615, 2426894, 3592164, 5817845, 8360196, 10396523, 21941765, 33649653, 48804040, 61413482, 124029358

Offset: 0

Labos Elemer, Apr 25 2001

nonn

			For n up to 7 all divisor differences of n! are also divisors of n!.
For n = 8, there are 3 divisor differences of 8! = 40320 which are not divisors of 8!, namely 27, 54 and 108.

Cf. A000142, A027423, A060737, A060742, A060763.

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

a(n) = {my(v = List(), f = n!, d1 = 1, del); fordiv(f, d, if(d > 1, del = d - d1; if(f % del, listput(v, del)); d1 = d)); #Set(v);} \\ Amiram Eldar, Jun 15 2024

a(n) = A060763(n!).

Edited by Dean Hickerson, Jan 22 2002
More terms from Sean A. Irvine, Dec 21 2022
a(41) from Amiram Eldar, Jun 15 2024

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).

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

A060654 a(n) = gcd(n, A060766(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 5, 13, 9, 14, 1, 30, 1, 16, 11, 17, 7, 18, 1, 19, 13, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 11, 28, 19, 29, 1, 60, 1, 31, 21, 32, 13, 33, 1, 34, 23, 70, 1, 36, 1, 37, 25, 38, 11, 39, 1, 40, 27, 41

Offset: 2

Labos Elemer, Apr 25 2001

nonn

			If n is prime p, then A060766(p) = p-1 and lcm(p, p-1) = 1. If n=2k then a(2k)=k or as an "anomaly", a(2k)=2k.
At n=30, D={1, 2, 3, 5, 6, 10, 15, 30}, dD={1, 1, 2, 1, 4, 5, 15}={1, 2, 4, 5, 15}, lcm(dD)=60, gcd(n, lcm(dD(n))) = gcd(30, 60) = 30 = n.
At n=36, D={1, 2, 3, 4, 6, 9, 12, 18, 36}, dD={1, 1, 1, 2, 3, 3, 6, 18}={1, 2, 3, 6, 18}, lcm(dD)=18, gcd(n, lcm(dD(n))) = gcd(36, 18) = 18 = n/2.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000005, A060680-A060685, A060741, A060742, A060763-A060766.

A060766:= proc(n) local F; F:= sort(convert(numtheory:-divisors(n),list));
   ilcm(op(F[2..-1] - F[1..-2])) end proc:
seq(igcd(n,A060766(n)),n=2..100); # Robert Israel, Dec 20 2015

Table[GCD[n, LCM @@ Differences@ Divisors@ n], {n, 2, 82}] (* Michael De Vlieger, Dec 20 2015 *)

a(n) = gcd(n, lcm(dd(n))), where dd(n) is the first difference of divisors (ordered by size).

A060695 a(n) = gcd(2n, A060766(2n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 30, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 60, 31, 32, 33, 34, 70, 36, 37, 38, 39, 40, 41, 42, 43, 44, 90, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 126, 64, 65, 66, 67, 68, 69, 140, 71

Offset: 1

Labos Elemer, Apr 25 2001

nonn

			n = 30: D = {1, 2, 3, 5, 6, 10, 15, 30}, dD = {1, 1, 2, 1, 4, 5, 15}={1, 2, 4, 5, 15}, lcm(dD) = 60, gcd(n, lcm(dD(n))) = gcd(30, 60) = 30 = n
n = 36: D = {1, 2, 3, 4, 6, 9, 12, 18, 36}, dD = {1, 1, 1, 2, 3, 3, 6, 18} = {1, 2, 3, 6, 18}, lcm(dD) = 18, gcd(n, lcm(dD(n))) = gcd(36, 18) = 18 = n/2.

Cf. A000005, A060680-A060685, A060741, A060742, A060763-A060766.

Table[GCD[2 n, LCM @@ Differences@ Divisors[2 n]], {n, 71}] (* Michael De Vlieger, Dec 20 2015 *)

a(n) = my(d=divisors(2*n), dd = vector(#d-1, k, d[k+1] - d[k])); gcd(2*n, lcm(dd)); \\ Michel Marcus, Mar 22 2020

a(n) = a(2k) is either n = 2k or n/2 = k. a(n) = n/2 seems regular, a(n) = n seems "anomalous".

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

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A060764 Number of divisors of n which are not also differences between consecutive divisors (ordered by increasing magnitude) of n.

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A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.

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A060738 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n! which are not also divisors of n!.

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

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A060654 a(n) = gcd(n, A060766(n)).

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A060695 a(n) = gcd(2n, A060766(2n)).

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