A328024 -id:A328024 - 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

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).

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}]

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A328026 Number of divisible pairs of consecutive divisors of n.

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

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

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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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