A060680 -id:A060680 - cp's OEIS frontend

A328025 Irregular triangle read by rows where row n gives the differences between consecutive divisors of n in weakly decreasing order.

1, 2, 2, 1, 4, 3, 1, 1, 6, 4, 2, 1, 6, 2, 5, 3, 1, 10, 6, 2, 1, 1, 1, 12, 7, 5, 1, 10, 2, 2, 8, 4, 2, 1, 16, 9, 3, 3, 1, 1, 18, 10, 5, 2, 1, 1, 14, 4, 2, 11, 9, 1, 22, 12, 4, 2, 2, 1, 1, 1, 20, 4, 13, 11, 1, 18, 6, 2, 14, 7, 3, 2, 1, 28, 15, 5, 4, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Oct 02 2019

			Triangle begins:
   {}
   1
   2
   2  1
   4
   3  1  1
   6
   4  2  1
   6  2
   5  3  1
  10
   6  2  1  1  1
  12
   7  5  1
  10  2  2
   8  4  2  1
  16
   9  3  3  1  1
  18
  10  5  2  1  1
For example, the divisors of 18 are {1,2,3,6,9,18}, with differences {1,1,3,3,9}, so row 18 is {9,3,3,1,1}.

Same as A193829 with rows sorted in weakly decreasing order.
Same as A328027 with rows reversed.
Row sums are A001477.
Row lengths are A000005.
First column is A060681.
Heinz numbers of rows are A328023.
Cf. A027750, A060680, A060682, A060683, A070211, A129308, A328024.

Table[Sort[Differences[Divisors[n]],Greater],{n,30}]

A328027 Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 1, 3, 6, 1, 2, 4, 2, 6, 1, 3, 5, 10, 1, 1, 1, 2, 6, 12, 1, 5, 7, 2, 2, 10, 1, 2, 4, 8, 16, 1, 1, 3, 3, 9, 18, 1, 1, 2, 5, 10, 2, 4, 14, 1, 9, 11, 22, 1, 1, 1, 2, 2, 4, 12, 4, 20, 1, 11, 13, 2, 6, 18, 1, 2, 3, 7, 14, 28, 1, 1, 1, 2, 4, 5, 15

Offset: 1

Gus Wiseman, Oct 02 2019

			Triangle begins:
   {}
   1
   2
   1  2
   4
   1  1  3
   6
   1  2  4
   2  6
   1  3  5
  10
   1  1  1  2  6
  12
   1  5  7
   2  2 10
   1  2  4  8
  16
   1  1  3  3  9
  18
   1  1  2  5 10
   2  4 14
   1  9 11
  22
   1  1  1  2  2  4 12
For example, the divisors of 18 are {1,2,3,6,9,18}, with differences {1,1,3,3,9}, which is row 18.

Same as A193829 with rows sorted in increasing order.
Same as A328025 with rows reversed.
Row sums are A001477.
Row lengths are A000005.
First column is A060680.
Cf. A027750, A060681, A060682, A060683, A070211, A129308, A328024.

Table[Sort[Differences[Divisors[n]]],{n,30}]

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).

A214055 Least m>0 such that n!+2+m and n-m have a common divisor > 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 2, 9, 1, 11, 2, 1, 2, 15, 1, 17, 2, 1, 2, 21, 1, 3, 2, 1, 2, 27, 1, 29, 2, 1, 2, 3, 1, 35, 2, 1, 2, 39, 1, 41, 2, 1, 2, 45, 1, 5, 2, 1, 2, 51, 1, 3, 2, 1, 2, 57, 1, 59, 2, 1, 2, 3, 1, 65, 2, 1, 2, 69, 1, 71, 2, 1, 2, 5, 1, 77, 2, 1, 2

Offset: 1

Clark Kimberling, Jul 22 2012

			6!+2+1=723, 6-1=5; 6!+2+2=724, 6-2=4; so a(6)=2.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A020639, A060680.

b[n_] := n! + 2; c[n_] := n; Table[m = 1; While[GCD[b[n] + m, c[n] - m] == 1, m++]; m, {n, 100}]

A253266 Numbers such that the minimum distance between divisors of n occurs only between composite numbers.

Original entry on oeis.org

14399, 34595, 82943, 89999, 100793, 116963, 158389, 172975, 224675, 244783, 245021, 266255, 278783, 281957, 285155, 304703, 331177, 338723, 343387, 380545, 417571, 446369, 447557, 484415, 497021, 532763, 580601, 585221, 588115, 590359, 608399, 619157, 627263, 629993

Offset: 1

Franklin T. Adams-Watters, May 01 2015

nonn

All members of the sequence are either a square minus a small square, or a multiple of a smaller member of the sequence. The square root of the small square must be less than half the fourth root of the number.

245021 is the smallest member of this sequence that is neither 1 less than a square nor a multiple of a smaller member of the sequence. It is 4 less than a square.

			The divisors of 14399 = 119*121 are 1, 7, 11, 17, 77, 119, 121, 187, 847, 1309, 2057, 14399. The minimum difference between divisors is 2, which occurs only between 119 and 121, both of which are composite; so 14399 is in the sequence.
The divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The minimum difference is 1, which does occur between two composites - 8, 9 - but it also occurs between pairs not both composite (e.g. 1, 2 or 3, 4), so 72 is not in the sequence.

Franklin T. Adams-Watters, Proof of assertion

Cf. A060680 (minimum difference), A033676 and A033677 (central divisors).

isa(n) = local(ds=divisors(n),diff,mind,dcomp);mind=n;for(k=2,#ds,diff=ds[k]-ds[k-1];if(diff<=mind,if(diff

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

A328025 Irregular triangle read by rows where row n gives the differences between consecutive divisors of n in weakly decreasing order.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A328027 Irregular triangle read by rows where row n lists, in weakly increasing order, the differences between consecutive divisors of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A214055 Least m>0 such that n!+2+m and n-m have a common divisor > 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A253266 Numbers such that the minimum distance between divisors of n occurs only between composite numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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