A060682 -id:A060682 - cp's OEIS frontend

A060681 Largest difference between consecutive divisors of n (ordered by size).

0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 6, 12, 7, 10, 8, 16, 9, 18, 10, 14, 11, 22, 12, 20, 13, 18, 14, 28, 15, 30, 16, 22, 17, 28, 18, 36, 19, 26, 20, 40, 21, 42, 22, 30, 23, 46, 24, 42, 25, 34, 26, 52, 27, 44, 28, 38, 29, 58, 30, 60, 31, 42, 32, 52, 33, 66, 34, 46, 35, 70, 36, 72, 37

Offset: 1

Labos Elemer, Apr 19 2001

Is a(n) the least m > 0 such that n - m divides n! + m? - Clark Kimberling, Jul 28 2012
Is a(n) the least m > 0 such that L(n-m) divides L(n+m), where L = A000032 (Lucas numbers)? - Clark Kimberling, Jul 30 2012
Records give A006093. - Omar E. Pol, Oct 26 2013
Divide n by its smallest prime factor p, then multiply with (p-1), with a(1) = 0 by convention. Compare also to A366387. - Antti Karttunen, Oct 23 2023
a(n) is also the smallest LCM of positive integers x and y where x + y = n. - Felix Huber, Aug 28 2024

			For n = 35, divisors are {1, 5, 7, 35}; differences are {4, 2, 28}; a(35) = largest difference = 28 = 35 - 35/5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Antal Balog, Paul Erdős, and Gérald Tenenbaum,, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Cf. A013661, A020639, A060680, A060682, A060683, A060685, A064097 (number of iterations needed to reach 1).

Cf. also A171462, A366387.

a060681 n = div n p * (p - 1) where p = a020639 n
-- Reinhard Zumkeller, Apr 06 2015

read("transforms") :
A060681 := proc(n)
    if n = 1 then
        0 ;
    else
        sort(convert(numtheory[divisors](n),list)) ;
        DIFF(%) ;
        max(op(%)) ;
    end if;
end proc:
seq(A060681(n),n=1..60) ; # R. J. Mathar, May 23 2018
# second Maple program:
A060681:=n->if(n=1,0,min(map(x->ilcm(x,n-x),[$1..1/2*n]))); seq(A060681(n),n=1..74); # Felix Huber, Aug 28 2024

a[n_ ] := n - n/FactorInteger[n][[1, 1]]
Array[Max[Differences[Divisors[#]]] &, 80, 2] (* Harvey P. Dale, Oct 26 2013 *)

diff(v)=vector(#v-1,i,v[i+1]-v[i])
a(n)=vecmax(diff(divisors(n))) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = if (n==1, 0, n - n/factor(n)[1,1]); \\ Michel Marcus, Oct 24 2015

first(n) = n = max(n, 1); my(res = vector(n)); res[1] = 0; forprime(p = 2, n, for(i = 1, n \ p, if(res[p * i] == 0, res[p * i] = i*(p-1)))); res \\ David A. Corneth, Jan 08 2019

from sympy import primefactors
def A060681(n): return n-n//min(primefactors(n),default=1) # Chai Wah Wu, Jun 21 2023

a(n) = n - n/A020639(n).
a(n) = n - A032742(n). - Omar E. Pol, Aug 31 2011
a(2n) = n, a(3*(2n+1)) = 2*(2n+1) = 4n + 2. - Antti Karttunen, Oct 23 2023
Sum_{k=1..n} a(k) ~ (1/2 - c) * n^2, where c is defined in the corresponding formula in A032742. - Amiram Eldar, Dec 21 2024

Edited by Dean Hickerson, Jan 22 2002

a(1)=0 added by N. J. A. Sloane, Oct 01 2015 at the suggestion of Antti Karttunen

A060680 Smallest difference between consecutive divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 2, 1, 22, 1, 4, 1, 2, 1, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 2, 1, 46, 1, 6, 1, 2, 1, 52, 1, 4, 1, 2, 1, 58, 1, 60, 1, 2, 1, 4, 1, 66, 1, 2, 1, 70, 1, 72, 1, 2, 1, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 6, 1, 2, 1, 4

Offset: 2

Labos Elemer, Apr 19 2001

nonn

a(n) = 1 if n is even and a(n) is even if n is odd.

a(n) = least m>0 such that n!+1+m and n-m are not relatively prime. - Clark Kimberling, Jul 21 2012

			For n = 35, divisors = {1,5,7,35}; differences = {4,2,28}; a(35) = smallest difference = 2.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Antal Balog, Paul Erdős and Gérald Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Cf. A060681 (largest difference), A060682, A060683, A060684.

Cf. A193829, A027750.

a060680 = minimum . a193829_row  -- Reinhard Zumkeller, Jun 25 2015

read("transforms") :
A060680 := proc(n)
    sort(convert(numtheory[divisors](n),list)) ;
    DIFF(%) ;
    min(op(%)) ;
end proc:
seq(A060680(n),n=2..60) ; # R. J. Mathar, May 23 2018

a[n_] := Min@@(Drop[d=Divisors[n], 1]-Drop[d, -1]);
(* Second program: *)
a[n_] := Min[Differences[Divisors[n]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Oct 16 2024 *)

a(n) = {my(m = n, d1); fordiv(n, d, if(d > 1 && d - d1 < m, m = d - d1); d1 = d); m;} \\ Amiram Eldar, Mar 17 2025

a(2n+1) = A060684(n).

Corrected by David W. Wilson, May 04 2001

Edited by Dean Hickerson, Jan 22 2002

A193829 Irregular triangle read by rows in which row n lists 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, 2, 1, 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, 2, 1, 4, 5, 15, 30

Offset: 2

Omar E. Pol, Aug 31 2011

The sum of row n gives A000027(n-1). The product of row n gives A057449(n). Row n has length A032741(n). The final term of row n is A060681(n). It appears that the first term of row n is A057237(n).

			Written as a triangle:
1,
2,
1, 2,
4,
1, 1, 3,
6,
1, 2, 4,
2, 6,
1, 3, 5,
10,
1, 1, 1, 2, 6

Alois P. Heinz, Rows n = 2..1600, flattened
Omar E. Pol, Illustration of divisors of n
Wikipedia, Table of divisors

Cf. A027750, A032742.

Cf. A060682 (distinct terms per row), A060680 (row minima), A060681 (row maxima).

import Data.List (genericIndex)
a193829 n k = genericIndex a193829_tabf (n - 1) !! (k - 1)
a193829_row n = genericIndex a193829_tabf (n - 1)
a193829_tabf = zipWith (zipWith (-))
                       (map tail a027750_tabf') a027750_tabf'
-- Reinhard Zumkeller, Jun 25 2015, Jun 23 2013

Flatten[Table[Differences[Divisors[n]], {n, 2, 30}]] (* T. D. Noe, Aug 31 2011 *)

T(n,k) = A027750(n,k+1)-A027750(n,k). - R. J. Mathar, Sep 01 2011

A060683 Numbers for which the differences between consecutive divisors (ordered by size) are distinct.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98

Offset: 1

Labos Elemer, Apr 19 2001

nonn

A060682(a(n)) = A000005(a(n)) - 1, n > 1. - Reinhard Zumkeller, Jun 25 2015

			For n=6, divisors={1,2,3,6}; differences={1,1,3}, which are not distinct, so 6 is not in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Balog, P. Erdős, and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.

Cf. A000005, A060680, A060681, A060682, A060765.

Cf. A060682, A259366 (complement).

a060683 n = a060683_list !! (n-1)
a060683_list = 1 : filter (\x -> a060682 x == a000005' x - 1) [2..]
-- Reinhard Zumkeller, Jun 25 2015

test[n_ ] := Length[dd=Drop[d=Divisors[n], 1]-Drop[d, -1]]==Length[Union[dd]]; Select[Range[1, 100], test]

isok(k) = my(d=divisors(k)); #Set(vector(#d-1, k, d[k+1]-d[k])) == #d-1; \\ Michel Marcus, Nov 11 2023

Edited by Dean Hickerson, Jan 22 2002

A328026 Number of divisible pairs of consecutive divisors of n.

Original entry on oeis.org

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

A060763 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, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 3, 1, 1, 0, 2, 1, 3, 1, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 1, 1, 0, 2, 2, 3, 1, 1, 0, 3, 2, 3, 1, 1, 0, 1, 1, 4, 0, 3, 1, 1, 1, 3, 3, 1, 0, 1, 1, 3, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 0, 1, 2, 4, 0, 1, 1, 1, 1, 5

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 differences {3,4,21} are not divisors, so a(70)=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060682, A060738, A060764.

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:
map(f, [$1..200]); # Robert Israel, Jul 03 2017

a[n_] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]

a(n) = my(d=divisors(n)); #select(x->(setsearch(d, x)==0), vecsort(vector(#d-1, k, d[k+1] - d[k]),,8)); \\ Michel Marcus, Jul 04 2017

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).

A328024 Heinz numbers of multisets representing the differences between some positive integer's consecutive divisors.

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 20, 29, 37, 39, 42, 53, 61, 79, 107, 110, 113, 151, 173, 181, 199, 239, 261, 271, 281, 312, 317, 349, 359, 374, 397, 421, 457, 497, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 798, 857, 863, 903, 911, 953, 983, 1021, 1061, 1069, 1151

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).

There is exactly one entry with any given sum of prime indices A056239.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     3: {2}
     6: {1,2}
     7: {4}
    13: {6}
    20: {1,1,3}
    29: {10}
    37: {12}
    39: {2,6}
    42: {1,2,4}
    53: {16}
    61: {18}
    79: {22}
   107: {28}
   110: {1,3,5}
   113: {30}
   151: {36}
   173: {40}
   181: {42}
   199: {46}
   239: {52}
   261: {2,2,10}
   271: {58}
   281: {60}
   312: {1,1,1,2,6}
For example, the divisors of 8 are {1,2,4,8}, with differences {1,2,4}, with Heinz number 42, so 42 belongs to the sequence.

A permutation of A328023.
Also the set of possible Heinz numbers of rows of A193829, A328025, or A328027.
Cf. A001222, A000005, A027750, A056239, A060680, A060681, A060682, A112798.

nn=1000;
Select[Union[Table[Times@@Prime/@Differences[Divisors[n]],{n,nn}]],#<=nn&]

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}]

cp's OEIS Frontend

A060681 Largest difference between consecutive divisors of n (ordered by size).

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A060680 Smallest difference between consecutive divisors of n.

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A193829 Irregular triangle read by rows in which row n lists the differences between consecutive divisors of n.

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A060683 Numbers for which the differences between consecutive divisors (ordered by size) are distinct.

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A328026 Number of divisible pairs of consecutive divisors of n.

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A060763 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n which are not also divisors of n.

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