A060682 -id:A060682 - cp's OEIS frontend

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

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}]

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

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

A330757 Let d(1) < d(2) < ... < d(q) denote the divisors of n; a(n) is the number of elements of the set { d(1)/d(2), d(2)/d(3), ..., d(q-1)/d(q) }.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 6, 1, 2, 3, 1, 2, 3, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 1, 5, 2, 2, 2

Offset: 1

Rémy Sigrist, Dec 29 2019

nonn

This sequence is a variant of A060682; here we consider the quotients, there the differences of consecutive divisors.

The sequence is unbounded since a(n!) >= n-1 for any n > 0.

			For n = 42:
- the divisors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42,
- the corresponding quotients are: 1/2, 2/3, 1/2, 6/7, 1/2, 2/3, 1/2,
- which corresponds to the set { 1/2, 2/3, 6/7 },
- hence a(42) = 3.

Cf. A060682, A246655.

a(n) = my (d=divisors(n)); #Set(vector(#d-1, k, d[k]/d[k+1]))

a(n) = 1 iff n is a prime power (A246655).

cp's OEIS Frontend

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

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

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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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A330757 Let d(1) < d(2) < ... < d(q) denote the divisors of n; a(n) is the number of elements of the set { d(1)/d(2), d(2)/d(3), ..., d(q-1)/d(q) }.

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