A060683 -id:A060683 - cp's OEIS frontend

A328026 Number of divisible pairs of consecutive divisors of n.

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

Offset: 1

Gus Wiseman, Oct 03 2019

nonn

The number m = 2^n, n >= 0, is the smallest for which a(m) = n. - Marius A. Burtea, Nov 20 2019

			The divisors of 500 are {1,2,4,5,10,20,25,50,100,125,250,500}, with consecutive divisible pairs {1,2}, {2,4}, {5,10}, {10,20}, {25,50}, {50,100}, {125,250}, {250,500}, so a(500) = 8.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 1's are A000040.
Positions of 0's and 2's are A328028.
Positions of terms > 2 are A328189.
Successive pairs of consecutive divisors are counted by A129308.
Cf. A000005, A027750, A033676, A060680, A060681, A060682, A060683, A193829, A328023, A328024, A328194.

f:=func;  g:=func; [g(n):n in [1..100]]; // Marius A. Burtea, Nov 20 2019

Table[Length[Split[Divisors[n],!Divisible[#2,#1]&]]-1,{n,100}]

a(n) = {my(d=divisors(n), nb=0); for (i=2, #d, if ((d[i] % d[i-1]) == 0, nb++)); nb;} \\ Michel Marcus, Oct 05 2019

a(p^k) = k for any prime number p and k >= 0. - Rémy Sigrist, Oct 05 2019

Data section extended up to a(105) by Antti Karttunen, Feb 23 2023

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 13 2022

nonn

A sequence is gapless if it covers an unbroken interval of positive integers. For example, the multiset {2,3,5,5,6,9} has three maximal gapless submultisets: {2,3}, {5,5,6}, {9}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 18564 are {1,1,2,4,6,7}, with maximal gapless submultisets {1,1,2}, {4}, {6,7}, so a(18564) = 1.

Positions of first appearances are A000079.
The maximal gapless submultisets are counted by A287170, firsts A066205.
These are the row-minima of A356226, firsts A356232.
The greatest instead of smallest size is A356228.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A001223 lists the prime gaps, reduced A028334.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gapless prime indices, cf. A073492-A073495.
A356224 counts even gapless divisors, complement A356225.
Cf. A000005, A055874, A060680-A060683, A132747, A132881, A137921, A193829, A286470, A356229.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Min@@Length/@Split[primeMS[n],#1>=#2-1&]],{n,100}]

a(n) = A333768(A356230(n)).

a(n) = A055396(A356231(n)).

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

A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 4, 6, 15, 10, 6, 12, 35, 10, 8, 16, 9, 18, 10, 28, 99, 22, 12, 20, 143, 18, 42, 28, 60, 30, 16, 88, 255, 28, 18, 36, 323, 130, 60, 40, 21, 42, 154, 60, 483, 46, 24, 42, 75, 238, 234, 52, 27, 132, 84, 304, 783, 58, 60, 60, 899, 84, 32, 104, 165, 66, 442

Offset: 2

Labos Elemer, Apr 24 2001

nonn

			For n=98, divisors={1,2,7,14,49,98}; differences={1,5,7,35,49}; a(98) = LCM of differences = 245.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

The GCD version appears to be A258409.
The LCM of the prime indices of n is A290103(n).
The differences between consecutive divisors of n are row n of A193829.
Cf. A000005, A027750, A060680, A060681, A060683, A328023, A328026, A328219.

a[n_ ] := LCM@@(Drop[d=Divisors[n], 1]-Drop[d, -1])
Table[LCM@@Differences[Divisors[n]],{n,2,70}] (* Harvey P. Dale, Oct 08 2012 *)

a(n) = A290103(A328023(n)). - Gus Wiseman, Oct 16 2019

Edited by Dean Hickerson, Jan 22 2002

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

Original entry on oeis.org

1, 2, 3, 6, 7, 20, 13, 42, 39, 110, 29, 312, 37, 374, 261, 798, 53, 2300, 61, 3828, 903, 1426, 79, 18648, 497, 2542, 2379, 21930, 107, 86856, 113, 42294, 4503, 5546, 2247, 475800, 151, 7906, 8787, 370620, 173, 843880, 181, 249798, 92547, 12118, 199, 5965848

Offset: 1

Gus Wiseman, Oct 02 2019

nonn

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
            1: ()
            2: (1)
            3: (2)
            6: (2,1)
            7: (4)
           20: (3,1,1)
           13: (6)
           42: (4,2,1)
           39: (6,2)
          110: (5,3,1)
           29: (10)
          312: (6,2,1,1,1)
           37: (12)
          374: (7,5,1)
          261: (10,2,2)
          798: (8,4,2,1)
           53: (16)
         2300: (9,3,3,1,1)
           61: (18)
         3828: (10,5,2,1,1)
For example, the divisors of 6 are {1,2,3,6}, with differences {1,1,3}, with Heinz number 20, so a(6) = 20.

The sorted version is A328024.
a(n) is the Heinz number of row n of A193829, A328025, or A328027.
Cf. A000005, A027750, A056239, A060682, A060683, A112798, A129308, A193829.

Table[Times@@Prime/@Differences[Divisors[n]],{n,100}]

A056239(a(n)) = n - 1. In words, the integer partition with Heinz number a(n) is an integer partition of n - 1.
A055396(a(n)) = A060680(n).
A061395(a(n)) = A060681(n).
A001221(a(n)) = A060682(n).
A001222(a(n)) = A000005(n).

A259366 Numbers for which the differences between consecutive divisors (ordered by size) are not distinct.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 156, 160, 162, 165, 168, 174, 180, 182, 186, 189, 192, 195, 198, 200

Offset: 1

Reinhard Zumkeller, Jun 25 2015

nonn

			.    n |  A193829(n,*)      |  A027750(n,*)         |
.  ----+--------------------+-----------------------+------------
.   10 |  {1,3,5}           |  {1,2,5,10}           |
.   11 |  {10}              |  {1,11}               |
.   12 |  {1,1,1,2,6}       |  {1,2,3,4,6,12}       |  a(2) = 12
.   13 |  {12}              |  {1,13}               |
.   14 |  {1,5,7}           |  {1,2,7,14}           |
.   15 |  {2,2,10}          |  {1,3,5,15}           |  a(3) = 15
.   16 |  {1,2,4,8}         |  {1,2,4,8,16}         |
.   17 |  {16}              |  {1,17}               |
.   18 |  {1,1,3,3,9}       |  {1,2,3,6,9,18}       |  a(4) = 18
.   19 |  {18}              |  {1,19}               |
.   20 |  {1,2,1,5,10}      |  {1,2,4,5,10,20}      |  a(5) = 20
.   21 |  {2,4,14}          |  {1,3,7,21}           |
.   22 |  {1,9,11}          |  {1,2,11,22}          |
.   23 |  {22}              |  {1,23}               |
.   24 |  {1,1,1,2,2,4,12}  |  {1,2,3,4,6,8,12,24}  |  a(6) = 24
.   25 |  {4,20}            |  {1,5,25}             |            .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A193829, A027750, A060682, A000005, A060683 (complement), subsequence of A129512.

a259366 n = a259366_list !! (n-1)
a259366_list = filter (\x -> a060682 x < a000005' x - 1) [2..]

q[k_] := Module[{d = Differences[Divisors[k]]}, CountDistinct[d] < Length[d]]; Select[Range[200], q] (* Amiram Eldar, Jan 27 2025 *)

A060682(a(n)) < A000005(a(n)) - 1.

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

Original entry on oeis.org

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}]

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

A328026 Number of divisible pairs of consecutive divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A356227 Smallest size of a maximal gapless submultiset of the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Extensions

A060766 Least common multiple of differences between consecutive divisors of n (ordered by size).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A328023 Heinz number of the multiset of differences between consecutive divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A259366 Numbers for which the differences between consecutive divisors (ordered by size) are not distinct.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

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