A027750 -id:A027750 - cp's OEIS frontend

A060680 Smallest difference between consecutive divisors of n.

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

A261699 Triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists positive terms interleaved with k-1 zeros, starting in row k(k+1)/2. If k is odd the positive terms of column k are k's, otherwise if k is even the positive terms of column k are the odd numbers greater than k in increasing order.

Original entry on oeis.org

1, 1, 1, 3, 1, 0, 1, 5, 1, 0, 3, 1, 7, 0, 1, 0, 0, 1, 9, 3, 1, 0, 0, 5, 1, 11, 0, 0, 1, 0, 3, 0, 1, 13, 0, 0, 1, 0, 0, 7, 1, 15, 3, 0, 5, 1, 0, 0, 0, 0, 1, 17, 0, 0, 0, 1, 0, 3, 9, 0, 1, 19, 0, 0, 0, 1, 0, 0, 0, 5, 1, 21, 3, 0, 0, 7, 1, 0, 0, 11, 0, 0, 1, 23, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 1, 25, 0, 0, 5, 0, 1, 0, 0, 13, 0, 0

Offset: 1

Omar E. Pol, Sep 20 2015

Conjecture: the positive terms in row n are the odd divisors of n.
Note that the elements appear with an unusual ordering, for example; row 45 is 1, 45, 3, 0, 5, 15, 0, 0, 9.
The positive terms give A261697.
Row n has length A003056(n) hence column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
The sum of row n is A000593(n).
The connection with the symmetric representation of sigma is as follows: A237048 --> A235791 --> A237591 --> A237593.
Proof of the conjecture: let n = 2^m*s*t with s and t odd. The property stated in A237048 verifies the conjecture with odd divisor k <= A003056(n) of n in position k and odd divisor t > A003056(n) in position 2^(m+1)*s. Therefore reading in row n the nonzero odd positions from left to right and then the nonzero even positions from right to left gives all odd divisors of n in increasing order. - Hartmut F. W. Hoft, Oct 25 2015
A237048 gives the signum function (A057427) of this sequence. - Omar E. Pol, Nov 14 2016
From Peter Munn, Jul 30 2017: (Start)
Each odd divisor d of n corresponds to n written as a sum of consecutive integers (n/d - (d-1)/2) .. (n/d + (d-1)/2). After canceling any corresponding negative and positive terms and deleting any zero term, the lower bound becomes abs(n/d - d/2) + 1/2, leaving k terms where k = n/d + d/2 - abs(n/d - d/2). It can be shown T(n,k) = d.
This sequence thereby defines a one to one relationship between odd divisors of n and partitions of n into k consecutive parts.
The relationship is expressed below using 4 sequences (with matching row lengths), starting with this one:
A261699(n,k) = d, the odd divisor.
A211343(n,k) = abs(n/d - d/2) + 1/2, smallest part.
A285914(n,k) = k, number of parts.
A286013(n,k) = n/d + (d-1)/2, largest part.
If no partition of n into k consecutive parts exists, the corresponding sequence terms are 0.
(End)

			Triangle begins:
1;
1;
1,  3;
1,  0;
1,  5;
1,  0,  3;
1,  7,  0;
1,  0,  0;
1,  9,  3;
1,  0,  0,  5;
1, 11,  0,  0;
1,  0,  3,  0;
1, 13,  0,  0;
1,  0,  0,  7;
1, 15,  3,  0,  5;
1,  0,  0,  0,  0;
1, 17,  0,  0,  0;
1,  0,  3,  9,  0;
1, 19,  0,  0,  0;
1,  0,  0,  0,  5;
1, 21,  3,  0,  0,  7;
1,  0,  0, 11,  0,  0;
1, 23,  0,  0,  0,  0;
1,  0,  3,  0,  0,  0;
1, 25,  0,  0,  5,  0;
1,  0,  0, 13,  0,  0;
1, 27,  3,  0,  0,  9;
1,  0,  0,  0,  0,  0,  7;
...
From _Omar E. Pol_, Dec 19 2016: (Start)
Illustration of initial terms in a right triangle whose structure is the same as the structure of A237591:
Row                                                         _
1                                                         _|1|
2                                                       _|1 _|
3                                                     _|1  |3|
4                                                   _|1   _|0|
5                                                 _|1    |5 _|
6                                               _|1     _|0|3|
7                                             _|1      |7  |0|
8                                           _|1       _|0 _|0|
9                                         _|1        |9  |3 _|
10                                      _|1         _|0  |0|5|
11                                    _|1          |11  _|0|0|
12                                  _|1           _|0  |3  |0|
13                                _|1            |13   |0 _|0|
14                              _|1             _|0   _|0|7 _|
15                            _|1              |15   |3  |0|5|
16                          _|1               _|0    |0  |0|0|
17                        _|1                |17    _|0 _|0|0|
18                      _|1                 _|0    |3  |9  |0|
19                    _|1                  |19     |0  |0 _|0|
20                  _|1                   _|0     _|0  |0|5 _|
21                _|1                    |21     |3   _|0|0|7|
22              _|1                     _|0      |0  |11 |0|0|
23            _|1                      |23      _|0  |0  |0|0|
24          _|1                       _|0      |3    |0 _|0|0|
25        _|1                        |25       |0   _|0|5  |0|
26      _|1                         _|0       _|0  |13 |0 _|0|
27    _|1                          |27       |3    |0  |0|9 _|
28   |1                            |0        |0    |0  |0|0|7|
... (End)

Cf. A000217, A000593, A001227, A003056, A005408, A027750, A057427, A182469, A196020, A211343, A236104, A235791, A236112, A237048, A237591, A237593, A261350, A261697, A261698, A285914, A286013.

T[n_, k_?OddQ] /; n == k (k + 1)/2 := k; T[n_, k_?OddQ] /; Mod[n - k (k + 1)/2, k] == 0 := k; T[n_, k_?EvenQ] /; n == k (k + 1)/2 := k + 1; T[n_, k_?EvenQ] /; Mod[n - k (k + 1)/2, k] == 0 := T[n - k, k] + 2; T[, ] = 0; Table[T[n, k], {n, 1, 26}, {k, 1, Floor[(Sqrt[1 + 8 n] - 1)/2]}] // Flatten (* Jean-François Alcover, Sep 21 2015 *)
(* alternate definition using function a237048 *)
T[n_, k_] := If[a237048[n, k] == 1, If[OddQ[k], k, 2n/k], 0] (* Hartmut F. W. Hoft, Oct 25 2015 *)

From Hartmut F. W. Hoft, Oct 25 2015: (Start)
T(n, k) = 2n/k, if A237048(n, k) = 1 and k even,
and in accordance with the definition:
T(n, k) = k, if A237048(n, k) = 1 and k odd,
T(n, k) = 0 otherwise; for k <= A003056(n).
(End)
For m >= 1, d >= 1 and odd, T(m*d, m + d/2 - abs(m - d/2)) = d. - Peter Munn, Jul 24 2017

A007978 Least non-divisor of n.

Original entry on oeis.org

2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3

Offset: 1

Jeffrey Shallit

nonn

Least k >= 2 such that sigma(n) divides sigma(n*k), where sigma is A000203. - Benoit Cloitre, Dec 01 2002

Contains all and only the prime powers p^k, k > 0. The first occurrence of p^k is at A003418(p^k-1); so new records occur at indices in A051451. - Franklin T. Adams-Watters, Jun 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bakir Farhi, On the average asymptotic behavior of a certain type of sequences of integers, Integers, Vol. 9 (2009), pp. 555-567.

Cf. A003418, A051451, A055874, A053669, A061853, A027750, A064859.

Cf. also A266620 (least non-divisor of n!).

import Data.List ((\\))
a007978 = head . ([1..] \\) . a027750_row
-- Reinhard Zumkeller, May 10 2014

a:= proc(n) local k;
for k from 2 while n mod k = 0 do od:
k
end proc:
seq(a(n),n=1..100); # Robert Israel, Sep 02 2014

Table[k := 1; While[Mod[n, k] == 0, k++]; k, {n, 2000}]  (* Clark Kimberling, Jun 16 2012 *)
Join[{2, 3}, Table[Complement[Range[n], Divisors[n]][[1]], {n, 3, 100}]] (* Alonso del Arte, Sep 23 2017 *)

a(n) = {my(k=2); while(!(n % k), k++); k;} \\ Michel Marcus, Sep 25 2017

def a(n):
    k = 2
    while not n%k: k += 1
    return k
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 09 2022

def A007978(n): return next(filter(lambda d:n%d,range(2,n))) if n>2 else n+1 # Chai Wah Wu, Feb 22 2023

a(n) = A053669(n) + A061853(n) = A055874(n) + 1. - Henry Bottomley, May 10 2001
G.f.: sum(k >= 2, -k*(x^A003418(k) - x^A003418(k-1))/((x^A003418(k) - 1)*(x^A003418(k-1) - 1))). - Robert Israel, Sep 02 2014
From Alonso del Arte, Sep 23 2017: (Start)
a(n) < n for all n > 2.
a(2n + 1) = 2, a(2n) >= 3.
a(2^k) = 3 for k > 0.
a(n!) = prime(pi(n) + 1) for n >= 0, except for a(3!) = 4. (End)
Asymptotic mean: lim_{n->oo} Sum_{k=1..n} a(k) = 1 + A064859 (Farhi, 2009). - Amiram Eldar, Jun 29 2021

A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 14 2020

In other words: in row n replace every term of n-th row of A176206 with its divisors.
The terms in row n are also all parts of all partitions of n.
As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
From Omar E. Pol, Aug 01 2021: (Start)
The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For the connection with the tower described in A221529 see also A340035. (End)

			Triangle begins:
  [1];
  [1,2],   [1];
  [1,3],   [1,2],   [1],   [1];
  [1,2,4], [1,3],   [1,2], [1,2], [1],   [1],   [1];
  [1,5],   [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
  ...
For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  [1],
  -------
  [1, 2],
  [1],
  -------
  [1, 3],
  [1, 2],
  [1],
  [1];
  ----------
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1];
  ----------
  [1, 5],
  [1, 2, 4],
  [1, 3],
  [1, 3],
  [1, 2],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1],
  [1],
  [1];
.
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
.
|---|---------|-----|-------|---------|------------|---------------|
| n |         |  1  |   2   |    3    |      4     |       5       |
|---|---------|-----|-------|---------|------------|---------------|
| P |         |     |       |         |            |               |
| A |         |     |       |         |            |               |
| R |         |     |       |         |            |               |
| T |         |     |       |         |            |  5            |
| I |         |     |       |         |            |  3  2         |
| T |         |     |       |         |  4         |  4  1         |
| I |         |     |       |         |  2  2      |  2  2  1      |
| O |         |     |       |  3      |  3  1      |  3  1  1      |
| N |         |     |  2    |  2 1    |  2  1 1    |  2  1  1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1  1 1 1  |  1  1  1 1 1  |
----|---------|-----|-------|---------|------------|---------------|
.
|---|---------|-----|-------|---------|------------|---------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12  5 2 1  | 20  8  4 2 1  |
|   |         |  |  |  |/|  |  |/|/|  |  |/ |/|/|  |  |/ | /|/|/|  |
| L | A066633 |  1  |  2 1  |  4 1 1  |  7  3 1 1  | 12  4  2 1 1  |
| I |         |  *  |  * *  |  * * *  |  *  * * *  |  *  *  * * *  |
| N | A002260 |  1  |  1 2  |  1 2 3  |  1  2 3 4  |  1  2  3 4 5  |
| K |         |  =  |  = =  |  = = =  |  =  = = =  |  =  =  = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7  6 3 4  | 12  8  6 4 5  |
|   |         |  |  |  |\|  |  |\|\|  |  |\ |\|\|  |  |\ |\ |\|\|  |
|   | A206561 |  1  |  4 2  |  9 5 3  | 20 13 7 4  | 35 23 15 9 5  |
|---|---------|-----|-------|---------|------------|---------------|
.
|---|---------|-----|-------|---------|------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1  2   4  |  1         5  |
|   |---------|-----|-------|---------|------------|---------------|
|   | A027750 |     |  1    |  1 2    |  1    3    |  1  2    4    |
|   |---------|-----|-------|---------|------------|---------------|
| D | A027750 |     |       |  1      |  1  2      |  1     3      |
| I | A027750 |     |       |  1      |  1  2      |  1     3      |
| V |---------|-----|-------|---------|------------|---------------|
| I | A027750 |     |       |         |  1         |  1  2         |
| S | A027750 |     |       |         |  1         |  1  2         |
| O | A027750 |     |       |         |  1         |  1  2         |
| R |---------|-----|-------|---------|------------|---------------|
| S | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|---|---------|-----|-------|---------|------------|---------------|
.
Note that every row in the lower zone lists A027750.
Also the lower zone for every positive integer can be constructed using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5]) then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

Paolo Xausa, Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)

Nonzero terms of A340031.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812 (a subsequence).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.

A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
A338156[10] (* Generates 10 rows *) (* Paolo Xausa, Jan 12 2023 *)

A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
A338156(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 3, 2, 1, 7, 1, 8, 4, 2, 1, 9, 3, 1, 10, 5, 2, 1, 11, 1, 12, 6, 4, 3, 2, 1, 13, 1, 14, 7, 2, 1, 15, 5, 3, 1, 16, 8, 4, 2, 1, 17, 1, 18, 9, 6, 3, 2, 1, 19, 1, 20, 10, 5, 4, 2, 1, 21, 7, 3, 1, 22, 11, 2, 1, 23, 1, 24, 12, 8, 6, 4, 3, 2, 1, 25, 5, 1, 26, 13, 2, 1, 27, 9

Offset: 1

Antti Karttunen, Jun 20 2000

Old name was "Replace n by its divisors in reverse order."
This gives the second elements of the ordered pairs (a,b), a >= 1, b >= 1, ordered by their product ab.
T(n,k) = n / A027750(n,k) = A027750(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014
The 2nd column of the triangle is the largest proper divisor (A032742). - Charles Kusniec, Jan 30 2021

			Triangle begins:
1;
2, 1;
3, 1;
4, 2, 1;
5, 1;
6, 3, 2, 1;
7, 1;
8, 4, 2, 1;
9, 3, 1;
10, 5, 2, 1;
11, 1;
12, 6, 4, 3, 2, 1;
13, 1;
14, 7, 2, 1;
15, 5, 3, 1;
16, 8, 4, 2, 1;
17, 1;
18, 9, 6, 3, 2, 1;
19, 1;
20, 10, 5, 4, 2, 1;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Omar E. Pol, Illustration of the divisors of n - _Omar E. Pol_, Nov 22 2009

Cf. A027750 for the first elements, A056534, A168017, A000005 (row lengths), A000203 (row sums), A032742 (largest proper divisor).

a056538 n k = a056538_tabf !! (n-1) !! (k-1)
a056538_row n = a056538_tabf !! (n-1)
a056538_tabf = map reverse a027750_tabf
-- Reinhard Zumkeller, Sep 28 2014

Magma
```
[Reverse(Divisors(n)) : n in [1..30]];
```

map(op,[seq(reverse(sort(divisors(j))),j=1..30)]);
cdr := proc(l) if 0 = nops(l) then ([]) else (l[2..nops(l)]): fi: end:
reverse := proc(l) if 0 = nops(l) then ([]) else [op(reverse(cdr(l))), l[1]]; fi: end:

Table[Reverse@ Divisors@ n, {n, 27}] // Flatten (* Michael De Vlieger, Jul 27 2016 *)

row(n)=Vecrev(divisors(n)) \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A064894(A064896(n)).

Definition revised by N. J. A. Sloane, Jul 27 2016

A086971 Number of semiprime divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3

Offset: 1

Reinhard Zumkeller, Sep 22 2003

nonn

Inverse Moebius transform of A064911. - Jonathan Vos Post, Dec 08 2004

G. H. Hardy and E. M. Wright, Section 17.10 in An Introduction to the Theory of Numbers, 5th ed., Oxford, England: Clarendon Press, 1979.

T. D. Noe, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Mobius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, (1975), 789-803.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222, A220264 (least inverse).

a086971 = sum . map a064911 . a027750_row
-- Reinhard Zumkeller, Dec 14 2012

a:= proc(n) local l, m; l:=ifactors(n)[2]; m:=nops(l);
       m*(m-1)/2 +add(`if`(i[2]>1, 1, 0), i=l)
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jul 18 2013

semiPrimeQ[n_] := PrimeOmega@ n == 2; f[n_] := Length@ Select[Divisors@ n, semiPrimeQ@# &]; Array[f, 105] (* Zak Seidov, Mar 31 2011 and modified by Robert G. Wilson v, Dec 08 2012 *)
a[n_] := Count[e = FactorInteger[n][[;; , 2]], ?(# > 1 &)] + (o = Length[e])*(o - 1)/2; Array[a, 100] (* _Amiram Eldar, Jun 30 2022 *)

/* The following definitions of a(n) are equivalent. */
a(n) = sumdiv(n,d,bigomega(d)==2)
a(n) = f=factor(n); j=matsize(f)[1]; sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = f=factor(n); j=omega(n); sum(m=1,j,f[m,2]>=2) + binomial(j,2)
a(n) = omega(n/core(n)) + binomial(omega(n),2)
/* Rick L. Shepherd, Mar 06 2006 */

a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller, May 02 2005
a(n) = omega(n/core(n)) + binomial(omega(n),2) = A001221(n/A007913(n)) + binomial(A001221(n),2) = A056170(n) + A079275(n). - Rick L. Shepherd, Mar 06 2006
From Reinhard Zumkeller, Dec 14 2012: (Start)
a(n) = Sum_{k=1..A000005(n)} A064911(A027750(n,k)).
a(A220264(n)) = n and a(m) <> n for m < A220264(n); a(A008578(n)) = 0; a(A002808(n)) > 0; for n > 1: a(A102466(n)) <= 1 and a(A102467(n)) > 1; A066247(n) = A057427(a(n)). (End)
G.f.: Sum_{k = p*q, p prime, q prime} x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 25 2017

Entry revised by N. J. A. Sloane, Mar 28 2006

A033273 Number of nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6

Offset: 1

N. J. A. Sloane

nonn

Ray Chandler, Table of n, a(n) for n=1..100000

Cf. A000005, A001221, A010051, A027750.
Cf. A055079, A059992, A180040.
Cf. A001620, A077761.

a033273 = length . filter ((== 0) . a010051) . a027750_row
-- Reinhard Zumkeller, Dec 16 2013

[NumberOfDivisors(n) - #PrimeDivisors(n): n in [1..150]]; // Vincenzo Librandi, May 17 2017

Table[Length[Select[Divisors[n], ! PrimeQ[#] &]], {n, 104}] (* Jayanta Basu, May 23 2013 *)
Table[DivisorSigma[0, n] - PrimeNu[n], {n, 100}] (* Vincenzo Librandi, May 17 2017 *)
Table[Count[Divisors[n],?CompositeQ],{n,110}]+1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 11 2019 *)

a(n) = numdiv(n) - omega(n); \\ Michel Marcus, Apr 28 2016

a(n) = A000005(n) - A001221(n).
a((n)) = n and a(m) <> n for m < A055079(n). - Reinhard Zumkeller, Dec 16 2013
G.f.: Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Jan 17 2017
Dirichlet g.f.: zeta(s)*(zeta(s)-primezeta(s)). - Benedict W. J. Irwin, Jul 11 2018
Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 1 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Nov 27 2022

More terms from Reinhard Zumkeller, Sep 02 2003
Corrected error in offset. - Jaroslav Krizek, May 04 2009
Extended by Ray Chandler, Aug 07 2010

A161700 a(n) is the sum of the elements on the antidiagonal of the difference table of the divisors of n.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 13, 15, 19, 17, 21, 28, 25, 21, 41, 31, 33, 59, 37, 21, 53, 29, 45, 39, 61, 33, 65, 49, 57, 171, 61, 63, 77, 41, 117, 61, 73, 45, 89, -57, 81, 309, 85, 105, 167, 53, 93, -80, 127, 61, 113, 133, 105, 321, 173, 183, 125, 65, 117, -1039, 121, 69, 155, 127, 201, 333, 133, 189, 149, -69, 141, 117, 145, 81, 317, 217, 269

Offset: 1

Reinhard Zumkeller, Jun 17 2009, Jun 20 2009

sign

a(p^k) = p^(k+1) - (p-1)^(k+1) if p is prime. - Robert Israel, May 18 2016

			n=12: A000005(12)=6;
EDP(12,x) = (x^5 - 5*x^4 + 5*x^3 + 5*x^2 + 114*x + 120)/120 = A161701(x) is the interpolating polynomial for {(0,1),(1,2),(2,3),(3,4),(4,6),(5,12)},
{EDP(12,x): 0<=x<6} = {1, 2, 3, 4, 6, 12} = divisors of 12,
a(12) = EDP(12,6) = 28.
From _Peter Luschny_, May 18 2016: (Start)
a(40) = -57 because the sum of the elements on the antidiagonal of DTD(40) is -57.
The DTD(40) is:
[   1    2    4   5  8  10  20  40]
[   1    2    1   3  2  10  20   0]
[   1   -1    2  -1  8  10   0   0]
[  -2    3   -3   9  2   0   0   0]
[   5   -6   12  -7  0   0   0   0]
[ -11   18  -19   0  0   0   0   0]
[  29  -37    0   0  0   0   0   0]
[ -66    0    0   0  0   0   0   0]
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor
Eric Weisstein's World of Mathematics, Finite Difference
Reinhard Zumkeller, Enumerations of Divisors

Cf. A000012, A000027, A005408, A000124, A016813, A086514, A016921, A000125, A058331, A002522, A017281, A161701, A017533, A161702, A161703, A000127, A158057, A161704, A161705, A161706, A161707, A161708, A161709, A161710, A080856, A161711, A161712, A161713, A161714, A161715, A128470, A006261.

Cf. A161856.

f:= proc(n)
local D, nD;
D:= sort(convert(numtheory:-divisors(n),list));
nD:= nops(D);
CurveFitting:-PolynomialInterpolation([$0..nD-1],D, nD)
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2016

a[n_] := (d = Divisors[n]; t = Table[Differences[d, k], {k, 0, lg = Length[d]}]; Sum[t[[lg - k + 1, k]], {k, 1, lg}]);
Array[a, 77] (* Jean-François Alcover, Jan 25 2018 *)

def A161700(n):
    D = divisors(n)
    T = matrix(ZZ, len(D))
    for (m, d) in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
    return sum(T[k,len(D)-k-1] for k in range(len(D)))
print([A161700(n) for n in range(1,78)]) # Peter Luschny, May 18 2016

a(n) = EDP(n,tau(n)) with tau = A000005 and EDP(n,x) = interpolating polynomial for the divisors of n.
EDP(n,A000005(n) - 1) = n;
EDP(n,1) = A020639(n);
EDP(n,0) = 1;
EDP(n,k) = A027750(A006218(n-1)+k+1), 0<=k < A000005(n).

New name from Peter Luschny, May 18 2016

A243822 Number of k < n such that rad(k) | n but k does not divide n, where rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 2, 1, 0, 0, 4, 0, 2, 1, 3, 0, 3, 0, 3, 0, 2, 0, 10, 0, 0, 2, 4, 1, 5, 0, 4, 2, 3, 0, 11, 0, 3, 2, 4, 0, 5, 0, 6, 2, 3, 0, 8, 1, 3, 2, 4, 0, 14, 0, 4, 2, 0, 1, 14, 0, 4, 2, 12, 0, 6, 0, 5, 3, 4, 1, 15, 0, 4, 0, 5, 0, 16, 1, 5, 3, 3, 0, 20, 1, 4, 3, 5, 1, 8, 0, 7, 2, 6

Offset: 1

Michael De Vlieger, Jun 11 2014

nonn

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

			From _Michael De Vlieger_, Aug 11 2024: (Start)
Let S(n) = row n of A162306 and let D(n) = row n of A027750.a(2) = 0 since S(2) \ D(2) = {1, 2} \ {1, 2} is null.
a(10) = 2 since S(10) \ D(10) = {1, 2, 4, 5, 8, 10} \ {1, 2, 5, 10} = {4, 8}.a(16) = 0 since S(16) \ D(16) = {1, 2, 4, 8, 16} \ {1, 2, 4, 8, 16} is null, etc.Table of a(n) and S(n) \ D(n):
   n  a(n)  row n of A272618.
  ---------------------------
   6    1   {4}
  10    2   {4, 8}
  12    2   {8, 9}
  14    2   {4, 8}
  15    1   {9}
  18    4   {4, 8, 12*, 16}
  20    2   {8, 16}
  21    1   {9}
  22    3   {4, 8, 16}
  24    3   {9, 16, 18*}
  26    3   {4, 8, 16}
  28    2   {8, 16}
  30   10   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
Terms in A272618 marked with an asterisk are counted by A355432. All other terms are counted by A361235. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Exploring Number Bases as Tools, ACM Inroads, March 2012, Vol. 3, No. 1, pp. 4-12.
Michael De Vlieger, Regular and coregular numbers, ResearchGate, 2024.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20

Cf. A000005, A000961, A024619, A027750, A010846, A045763, A162306, A243823, A272618, A304570, A355432, A361235.

Table[Count[Range[n], ?(And[Divisible[n, Times @@ FactorInteger[#][[All, 1]]], ! Divisible[n, #]] &)], {n, 120}] (* _Michael De Vlieger, Aug 11 2024 *)

a(n) = A010846(n) - A000005(n) = card({row n of A162306} \ {row n of A027750}).
a(n) = A045763(n) - A243823(n).
a(n) = (Sum_{1<=k<=n, gcd(n,k)=1} mu(k)*floor(n/k)) - tau(n). - Michael De Vlieger, May 10 2016, after Benoit Cloitre at A010846.
From Michael De Vlieger, Aug 11 2024" (Start)
a(n) = 0 for n in A000961, a(n) > 0 for n in A024619.
a(n) = A051953(n) - A000005(n) + 1 = n - A000010(n) - A000005(n) - A243823(n) + 1.
a(n) = A355432(n) + A361235(n).
a(n) = A355432(n) for n in A360768.
a(n) = A361235(n) for n not in A360768.
a(n) = number of terms in row n of A272618.
a(n) = sum of row n of A304570. (End)

New name from David James Sycamore, Aug 11 2024

A137921 Number of divisors d of n such that d+1 is not a divisor of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 4, 4, 2, 4, 2, 4, 4, 3, 2, 5, 3, 3, 4, 5, 2, 5, 2, 5, 4, 3, 4, 6, 2, 3, 4, 6, 2, 5, 2, 5, 6, 3, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 3, 2, 7, 2, 3, 6, 6, 4, 6, 2, 5, 4, 7, 2, 8, 2, 3, 6, 5, 4, 6, 2, 8, 5, 3, 2, 8, 4, 3, 4, 7, 2, 8, 4, 5, 4, 3, 4, 9, 2, 5, 6, 7, 2, 6, 2, 7, 8

Offset: 1

Reinhard Zumkeller, Feb 23 2008

a(n) = number of "divisor islands" of n. A divisor island is any set of consecutive divisors of a number where no pairs of consecutive divisors in the set are separated by 2 or more. - Leroy Quet, Feb 07 2010

			The divisors of 30 are 1,2,3,5,6,10,15,30. The divisor islands are (1,2,3), (5,6), (10), (15), (30). (Note that the differences between consecutive divisors 5-3, 10-6, 15-10 and 30-15 are all > 1.) There are 5 such islands, so a(30)=5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.

Bisections: A099774, A174199.
First appearance of n is at position A173569(n).
Numbers whose divisors have no non-singleton runs are A005408.
The longest run of divisors of n has length A055874(n).
The number of successive pairs of divisors of n is A129308(n).
Cf. A000005, A001620, A027750, A060680, A088723, A088725, A181063, A199970, A328165, A328166, A328448, A328450.

a137921 n = length $ filter (> 0) $
   map ((mod n) . (+ 1)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 23 2011

with(numtheory): disl := proc (b) local ct, j: ct := 1: for j to nops(b)-1 do if 2 <= b[j+1]-b[j] then ct := ct+1 else end if end do: ct end proc: seq(disl(divisors(n)), n = 1 .. 120); # Emeric Deutsch, Feb 12 2010

f[n_] := Length@ Split[ Divisors@n, #2 - #1 == 1 &]; Array[f, 105] (* f(n) from Bobby R. Treat *) (* Robert G. Wilson v, Feb 22 2010 *)
Table[Count[Differences[Divisors[n]],?(#>1&)]+1,{n,110}] (* _Harvey P. Dale, Jun 05 2012 *)
a[n_] := DivisorSum[n, Boole[!Divisible[n, #+1]]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=my(d,s=0);if(n%2,numdiv(n),d=divisors(n);for(i=1,#d,if(n%(d[i]+1),s++));s)

a(n)=sumdiv(n,d,(n%(d+1)!=0)); \\ Joerg Arndt, Jan 06 2015

from sympy import divisors
def A137921(n):
    return len([d for d in divisors(n,generator=True) if n % (d+1)])
# Chai Wah Wu, Jan 05 2015

a(n) <= A000005(n), with equality iff n is odd; a(A137922(n)) = 2.
a(n) = A000005(n) - A129308(n). - Michel Marcus, Jan 06 2015
a(n) = A001222(A328166(n)). - Gus Wiseman, Oct 16 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 2), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

Corrected and edited by Charles R Greathouse IV, Apr 19 2010

Edited by N. J. A. Sloane, Aug 10 2010

cp's OEIS Frontend

A060680 Smallest difference between consecutive divisors of n.

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A007978 Least non-divisor of n.

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A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.

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A056538 Irregular triangle read by rows: row n lists the divisors of n in decreasing order.

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A086971 Number of semiprime divisors of n.

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