A060680 -id:A060680 - cp's OEIS frontend

A328189 Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).

8, 16, 18, 20, 27, 28, 32, 40, 42, 44, 50, 52, 54, 56, 64, 66, 68, 75, 76, 78, 80, 81, 88, 92, 98, 99, 100, 102, 104, 110, 112, 114, 116, 117, 124, 125, 126, 128, 130, 136, 138, 140, 147, 148, 152, 153, 156, 160, 162, 164, 170, 171, 172, 174, 176, 184, 186

Offset: 1

Gus Wiseman, Oct 13 2019

nonn

			The nontrivial divisors of 42 are {2, 3, 6, 7, 14, 21}, with pairs of consecutive divisible divisors {3, 6} and {7, 14}, so 42 belongs to the sequence.

Complement of A328161.
Positions of terms greater than 1 in A328194.
Partitions with a pair of consecutive divisible parts are A328221.
Cf. A000005, A060680, A060775, A067513, A088725, A163870, A328028, A328162, A328171.

Select[Range[200],MatchQ[DeleteCases[Divisors[#],1|#],{_,x_,y_,_}/;Divisible[y,x]]&]
Select[Range[2,200],AnyTrue[Partition[Most[Rest[Divisors[#]]],2,1],Mod[#[[2]],#[[1]]] == 0&]&] (* Harvey P. Dale, Mar 14 2023 *)

A328162 Maximum length of a divisibility chain of consecutive divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 4, 3, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 06 2019

nonn

			The divisors of 968 split into consecutive divisibility chains {{1, 2, 4, 8}, {11, 22, 44, 88}, {121, 242, 484, 968}}, so a(968) = 4.

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

Records occur at powers of 2 (A000079).
Taking only proper divisors gives A328194.
Taking only divisors > 1 gives A328195.
The maximum run-length among divisors of n is A055874.
Cf. A000005, A027750, A060680, A060775, A328026, A328028, A328161, A328189.

f:= proc(n) local F,L,d,i;
  F:= sort(convert(numtheory:-divisors(n),list));
  d:= nops(F);
  L:= Vector(d);
  L[1]:= 1;
  for i from 2 to d do
    if F[i] mod F[i-1] = 0 then L[i]:= L[i-1]+1
    else L[i]:= 1
    fi
  od;
  max(L)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 20 2023

Table[Max@@Length/@Split[Divisors[n],Divisible[#2,#1]&],{n,100}]

A328448 Smallest number whose divisors > 1 have a longest run of length n, and 0 if none exists.

Original entry on oeis.org

2, 6, 12, 504, 60, 420, 840, 4084080, 2520, 21162960, 27720, 2059318800, 0, 360360, 720720, 8494326640800, 12252240, 281206918792800, 0, 0, 232792560, 409547311252279200, 5354228880, 619808900849199341280, 26771144400, 54749786241679275146400, 80313433200, 5663770990518545704800

Offset: 1

Gus Wiseman, Oct 16 2019

nonn

			The runs of divisors of 504 (greater than 1) are {{2,3,4},{6,7,8,9},{12},{14},{18},{21},{24},{28},{36},{42},{56},{63},{72},{84},{126},{168},{252},{504}}, the longest of which has length 4, and 504 is the smallest number with this property, so a(4) = 504.

The version that looks at all divisors (including 1) is A328449.
The longest run of divisors of n greater than 1 has length A328457.
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A000005, A027750, A055874, A060680, A181063, A199970, A328450.

Data corrected and extended by Giovanni Resta, Oct 18 2019

A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

Original entry on oeis.org

0, 1, 2, 6, 12, 0, 60, 420, 840, 0, 2520, 0, 27720, 0, 0, 360360, 720720, 0, 12252240, 0, 0, 0, 232792560, 0, 5354228880, 0, 26771144400, 0, 80313433200, 0, 2329089562800, 72201776446800, 0, 0, 0, 0, 144403552893600, 0, 0, 0, 5342931457063200, 0

Offset: 0

Gus Wiseman, Oct 16 2019

nonn

Wikipedia, Run (cards)

Positions of 0's are 0 followed by A024619 - 1.
The version that looks only at all divisors > 1 is A328448.
The longest run of divisors of n has length A055874.
The longest run of divisors of n greater than one has length A328457.
Numbers whose divisors have no non-singleton runs are A005408.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a (not necessarily longest) maximal run of length n is A181063.
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A199970, A328195.

tav=Table[Max@@Length/@Split[Divisors[n],#2==#1+1&],{n,10000}];
Table[If[FreeQ[tav,i],0,Position[tav,i][[1,1]]],{i,0,Max@@tav}]

a(n) = LCM(1,2,...,n) = A003418(n) if n + 1 is a prime power, otherwise a(n) = 0.

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).

A328165 Numbers whose divisors do not have weakly decreasing run-lengths.

Original entry on oeis.org

56, 72, 110, 112, 132, 144, 156, 182, 210, 216, 224, 240, 264, 272, 288, 306, 312, 342, 364, 380, 392, 396, 420, 432, 440, 448, 462, 468, 480, 506, 528, 544, 550, 552, 576, 600, 612, 616, 624, 648, 650, 684, 702, 720, 728, 756, 760, 770, 780, 784, 792, 812

Offset: 1

Gus Wiseman, Oct 07 2019

nonn

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, with runs {{1, 2}, {4}, {7, 8}, {14}, {28}, {56}}, with lengths (2, 1, 2, 1, 1, 1), which are not weakly decreasing, so 56 is in the sequence.

Wikipedia, Run (cards)

The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The number of successive pairs of divisors of n is A129308(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A000005, A027750, A033676, A060680, A060775, A119313, A181063, A199970.

Select[Range[1000],!GreaterEqual@@Length/@Split[Divisors[#],#2==#1+1&]&]

A060684 Smallest difference between consecutive divisors (ordered by size) of 2n+1.

Original entry on oeis.org

2, 4, 6, 2, 10, 12, 2, 16, 18, 2, 22, 4, 2, 28, 30, 2, 2, 36, 2, 40, 42, 2, 46, 6, 2, 52, 4, 2, 58, 60, 2, 4, 66, 2, 70, 72, 2, 4, 78, 2, 82, 4, 2, 88, 6, 2, 4, 96, 2, 100, 102, 2, 106, 108, 2, 112, 4, 2, 6, 10, 2, 4, 126, 2, 130, 6, 2, 136, 138, 2, 2, 4, 2, 148, 150, 2, 4, 156, 2, 6

Offset: 1

Labos Elemer, Apr 19 2001

nonn

Successively greater values of a(n) occur when 2n+1 is prime.

			For n=38, 2n+1=77; divisors={1,7,11,77}; differences={6,4,66}; a(38) = smallest difference = 4.

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

Cf. A060680.
Different from A071294.
Cf. A193829, A027750.

a060684 = minimum . a193829_row . (+ 1) . (* 2)
-- Reinhard Zumkeller, Jun 25 2015

a[n_ ] := Min@@(Drop[d=Divisors[2n+1], 1]-Drop[d, -1])
Array[Min[Differences[Divisors[2*#+1]]]&,80] (* Harvey P. Dale, Dec 08 2013 *)

A060680(2n+1)

Edited by Dean Hickerson, Jan 22 2002

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&]

A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

Original entry on oeis.org

1, 2, 6, 12, 60, 72, 180, 360, 420, 840, 1260, 2520, 3780, 5040, 13860, 27720, 36960, 41580, 55440, 83160, 166320, 277200, 360360, 471240, 491400, 720720, 1081080, 1113840, 2162160, 2827440, 3341520, 4324320, 5405400, 6126120

Offset: 0

Gus Wiseman, Oct 15 2019

nonn

A sorted version of A287142.

			The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, with pairs of successive divisors {{1, 2}, {2, 3}, {3, 4}, {8, 9}}, and no smaller number has 4 successive pairs, so 72 belongs to the sequence.

Sorted positions of first appearances in A129308.
The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a longest run of length n is A328449(n).
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A287142, A328165.

dat=Table[Count[Differences[Divisors[n]],1],{n,10000}];
Sort[Table[Position[dat,i][[1,1]],{i,Union[dat]}]]

cp's OEIS Frontend

A328189 Numbers n with at least one pair of consecutive divisible nontrivial divisors (greater than 1 and less than n).

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A328162 Maximum length of a divisibility chain of consecutive divisors of n.

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A328448 Smallest number whose divisors > 1 have a longest run of length n, and 0 if none exists.

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A328449 Smallest number in whose divisors the longest run is of length n, and 0 if none exists.

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

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

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A328165 Numbers whose divisors do not have weakly decreasing run-lengths.

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A060684 Smallest difference between consecutive divisors (ordered by size) of 2n+1.

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Haskell

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

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