A060742 -id:A060742 - cp's OEIS frontend

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

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

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

A060737 Number of distinct differences between consecutive divisors of n! (ordered by size).

Original entry on oeis.org

0, 0, 1, 2, 4, 9, 15, 27, 44, 72, 116, 230, 379, 716, 1154, 1858, 2589, 5014, 7299, 15276, 21901, 33146, 52298, 102918, 136647, 200669, 327572, 432396, 596869, 1220172, 1603092, 3074018, 3752018, 5433507, 8555035, 11952469, 14590378, 30158257, 46318453, 66458379

Offset: 0

Labos Elemer, Apr 25 2001

nonn

			For n = 4, n! = 24; divisors = {1,2,3,4,6,8,12,24}; differences = {1,1,1,2,2,4,12}, distinct differences = {1,2,4,12}, so a(4) = 4.

Cf. A000142, A060682, A060738, A060742.

a[n_ ] := Length[Union[Drop[d=Divisors[n! ], 1]-Drop[d, -1]]]
Table[Length[Union[Differences[Divisors[n!]]]],{n,0,40}] (* Harvey P. Dale, Nov 22 2021 *)

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

a(n) = A060682(n!).

Edited by Dean Hickerson, Jan 22 2002
More terms from Ryan Propper, Mar 22 2006
a(37)-a(39) from Amiram Eldar, Jun 15 2024

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A060737 Number of distinct differences between consecutive divisors of n! (ordered by size).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI