A060741 -id:A060741 - cp's OEIS frontend

A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

0, 0, 1, 2, 4, 9, 15, 27, 41, 68, 111, 218, 328, 624, 929, 1518, 2016, 3689, 4965, 9252, 13177, 20016, 30697, 56749, 69434, 94242, 149558, 190292, 258370, 492924, 615063, 1149403, 1325124, 1841343, 2737190, 3592273, 4193855, 8216492, 12668800, 17654339, 20368544

Offset: 0

Labos Elemer, Apr 23 2001

nonn

			For n = 5, n! = 120; divisors = {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}; differences = {1,1,1,1,1,2,2,2,3,5,4,6,10,20,60}; intersection = {1,2,3,4,5,6,10,20,60}, so a(5) = 9.

Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.

Cf. A000142, A027423, A060737, A060738, A060741.

f:= proc(n) local D,L;
  D:= numtheory:-divisors(n!);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) intersect D);
end proc:
map(f, [$0..34]); # Robert Israel, Jul 03 2017

a[n_ ] := Length[Intersection[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) = A060741(n!/2) for n >= 2. - Amiram Eldar, Jun 15 2024

Edited by Dean Hickerson, Jan 22 2002
One more term from Robert G. Wilson v, Jan 29 2002
a(33)-a(35) from Robert Israel, Jul 03 2017
a(36)-a(40) from Amiram Eldar, Jun 15 2024

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

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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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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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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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