A027750 -id:A027750 - cp's OEIS frontend

A067513 Number of divisors d of n such that d+1 is prime.

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

Offset: 1

Amarnath Murthy, Feb 12 2002

1, 2 and 4 are the only numbers such that for every divisor d, d+1 is a prime.
These and only these primes appear as prime divisors of any term of InvPhi(n) set if n is not empty, i.e., if n is from A002202. - Labos Elemer, Jun 24 2002
a(n) is the number of integers k such that n = k - k/p where p is one of the prime divisors of k. (See, e.g., A064097 and A333123, which are related to the mapping k -> k - k/p.) - Robert G. Wilson v, Jun 12 2022

			a(12) = 5 as the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the corresponding primes are 2,3,5,7 and 13. Only 3+1 = 4 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty, arXiv:2208.06704 [math.NT], 2022.
Yuchen Ding, On a conjecture of R. M. Murty and V. K. Murty II, arXiv:2209.01087 [math.NT], 2022-2023.
Steve Fan and Carl Pomerance, Shifted-prime divisors, arXiv:2401.10427 [math.NT], 2024.
Mikhail R. Gabdullin, Moments of the shifted prime divisor function, arXiv preprint (2025). arXiv:2505.24050 [math.NT]
M. Ram Murty and V. Kumar Murty, A variant of the Hardy-Ramanujan theorem, Hardy-Ramanujan Journal, Vol. 44 (2021), pp. 32-40.
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Even-indexed terms give A046886.
Cf. A000005, A001221, A001222, A002202, A027750, A064097, A141197, A185633, A202727, A202728, A322312, A322976, A333123, A346467, A355452.
Cf. A005408 (positions of 1's), A051222 (of 2's).

a067513 = sum . map (a010051 . (+ 1)) . a027750_row
-- Reinhard Zumkeller, Jul 31 2012

A067513 := proc(n)
    local a,d;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isprime(d+1) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A067513(n),n=1..100) ; # R. J. Mathar, Aug 07 2022

a[n_] := Length[Select[Divisors[n]+1, PrimeQ]]
Table[Count[Divisors[n],?(PrimeQ[#+1]&)],{n,110}] (* _Harvey P. Dale, Feb 29 2012 *)
a[n_] := DivisorSum[n, 1 &, PrimeQ[# + 1] &]; Array[a, 100] (* Amiram Eldar, Jan 11 2025 *)

a(n)=sumdiv(n,d,isprime(d+1)) \\ Charles R Greathouse IV, Dec 23 2011

from sympy import divisors, isprime
def a(n): return sum(1 for d in divisors(n, generator=True) if isprime(d+1))
print([a(n) for n in range(1, 104)]) # Michael S. Branicky, Jul 12 2022

a(n) = 2 iff Bernoulli number B_{n} has denominator 6 (cf. A051222). - Vladeta Jovovic, Feb 13 2002
a(n) <= A141197(n). - Reinhard Zumkeller, Oct 06 2008
a(n) = A001221(A027760(n)). - Enrique Pérez Herrero, Dec 23 2011
a(n) = Sum_{k = 1..A000005(n)} A010051(A027750(n,k)+1). - Reinhard Zumkeller, Jul 31 2012
a(n) = A001221(A185633(n)) = A001222(A322312(n)). - Antti Karttunen, Jul 12 2022
Sum_{k=1..n} a(k) = n * (log(log(n)) + B) + O(n/log(n)), where B is a constant (Prachar, 1955). - Amiram Eldar, Jan 11 2025

Edited by Dean Hickerson, Feb 12 2002

A187202 The bottom entry in the difference table of the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 4, 0, 10, 1, 12, -2, 8, 1, 16, 12, 18, -11, 8, -6, 22, -12, 16, -8, 8, -3, 28, 50, 30, 1, 8, -12, 28, -11, 36, -14, 8, -66, 40, 104, 42, 13, 24, -18, 46, -103, 36, -16, 8, 21, 52, 88, 36, 48, 8, -24, 58, -667, 60, -26, -8, 1, 40, 72

Offset: 1

Omar E. Pol, Aug 01 2011

Note that if n is prime then a(n) = n - 1.
Note that if n is a power of 2 then a(n) = 1.
a(A193671(n)) > 0; a(A187204(n)) = 0; a(A193672(n)) < 0. [Reinhard Zumkeller, Aug 02 2011]
First differs from A187203 at a(14). - Omar E. Pol, May 14 2016
From David A. Corneth, May 20 2016: (Start)
The bottom of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
a
. . b-a
b . . . . c-2b+a
. . c-b . . . . . d-3c+3b-a
c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
. . d-c . . . . . e-3d+3c-b
d . . . . e-2d+c
. . e-d
e
From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
(End)

			a(18) = 12 because the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is:
1 . 2 . 3 . 6 . 9 . 18
. 1 . 1 . 3 . 3 . 9
. . 0 . 2 . 0 . 6
. . . 2 .-2 . 6
. . . .-4 . 8
. . . . . 12
with bottom entry a(18) = 12.
Note that A187203(18) = 4.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000005, A007318, A027750, A187203, A273102.

a187202 = head . head . dropWhile ((> 1) . length) . iterate diff . divs
   where divs n = filter ((== 0) . mod n) [1..n]
         diff xs = zipWith (-) (tail xs) xs
-- Reinhard Zumkeller, Aug 02 2011

f:=proc(n) local k,d,lis; lis:=divisors(n); d:=nops(lis);
add( (-1)^k*binomial(d-1,k)*lis[d-k], k=0..d-1); end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, May 01 2016

Table[d = Divisors[n]; Differences[d, Length[d] - 1][[1]], {n, 100}] (* T. D. Noe, Aug 01 2011 *)

A187202(n)={ for(i=2,#n=divisors(n), n=vecextract(n,"^1")-vecextract(n,"^-1")); n[1]}  \\ M. F. Hasler, Aug 01 2011

a(n) = Sum_{k=0..d-1} (-1)^k*binomial(d-1,k)*D[d-k], where D is a sorted list of the d = A000005(n) divisors of n. - N. J. A. Sloane, May 01 2016

a(2^k) = 1.

Edited by N. J. A. Sloane, May 01 2016

A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2020

Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section).

For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.

			Triangle begins:
  [1];
  [1, 2];
  [1, 3],       [1];
  [1, 2, 4],    [1, 2],    [1];
  [1, 5],       [1, 3],    [1, 2], [1],    [1];
  [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1];
  ...
For n = 6 the 6th row of A336811 is [6, 4, 3, 2, 2, 1, 1] so replacing every term with its divisors we have {[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]} the same as the 6th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  -------------
  [1],
  -------------
  [1, 2];
  -------------
  [1, 3],
  [1];
  -------------
  [1, 2, 4],
  [1, 2],
  [1];
  -------------
  [1, 5],
  [1, 3],
  [1, 2],
  [1],
  [1];
  -------------
  [1, 2, 3, 6],
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1];
  -------------
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and the parts of the last section of the set of partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the last section of the set of partitions of every positive integer.
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      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |    1          |
| I |         |     |       |         |           |  3 2        |      1        |
| T |         |     |       |         |  4        |    1        |      1        |
| I |         |     |       |         |  2 2      |      1      |        1      |
| O |         |     |       |  3      |    1      |      1      |        1      |
| N |         |     |  2    |    1    |      1    |        1    |          1    |
| S |         |  1  |    1  |      1  |        1  |          1  |            1  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A207031 |  1  |  2 1  |  3 1 1  |  6 3 1 1  |  8 3 2 1 1  | 15 8 4 2 1 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |  |/|/|/|/|/|  |
| I | A182703 |  1  |  1 1  |  2 0 1  |  3 2 0 1  |  5 1 1 0 1  |  7 4 2 1 0 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |  * * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |  1 2 3 4 5 6  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |  = = = = = =  |
|   | A207383 |  1  |  1 2  |  2 0 3  |  3 4 0 4  |  5 2 3 0 5  |  7 8 6 4 0 6  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
| D |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
| V |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S |---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
Note that every row in the lower zone lists A027750.
The "section" is the simpler substructure of the set of partitions of n that has this property in the three zones.
Also the lower zone for every positive integer can be constructed using the first n terms of A002865. For example: for n = 6 we consider the first 6 terms of A002865 (that is [1, 0, 1, 1, 2, 2]) and then the 6th slice is formed by a block with the divisors of 6, no block with the divisors of 5, one block with the divisors of 4, one block with the divisors of 3, two blocks with the divisors of 2 and two 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 growth step by step of both the prism of partitions and its associated tower since the number of parts in the last section of the set of partitions of n is equal to A138137(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 in the last section of the set of partitions of n is equal to A138879(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..10206 (rows 1..23 of triangle, flattened)

Companion and subsequence of A338156.
Row lengths give A138137.
Row sums give A138879.
Cf. A000041, A000070, A002260, A002865, A006128, A024916, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A187219, A207031, A207038, A207383, A221529, A221530, A221531, A237593, A245095, A221649, A221650, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A350357.

A336812[row_]:=Flatten[Table[ConstantArray[Divisors[row-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336812,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 16 2023 *)

A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 12, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 17, 1, 4, 7, 7, 4, 12, 1, 13, 4, 4

Offset: 1

Labos Elemer, Jul 19 2001

nonn

Let us say that two divisors d_1 and d_2 of n are adjacent divisors if d_1/d_2 or d_2/d_1 is a prime. Then a(n) is the number of all pairs of adjacent divisors of n. - Vladimir Shevelev, Aug 16 2010
Equivalent to the preceding comment: a(n) is the number of edges in the directed multigraph on tau(n) vertices, vertices labeled by the divisors d_i of n, where edges connect vertex(d_i) and vertex(d_j) if the ratio of the labels is a prime. - R. J. Mathar, Sep 23 2011
a(A001248(n)) = 2. - Reinhard Zumkeller, Dec 02 2014
Depends on the prime signature of n as follows: a(A025487(n)) = 0, 1, 2, 4, 3, 7, 4, 10, 12, 5, 12, 13, 20, 6, 17, 16, 28, 7, 22, 33, 19 ,32, 24, 36, 8, 27, 46, ... (n>=1). - R. J. Mathar, May 28 2017

			n = 255: divisors = {1, 3, 5, 15, 17, 51, 85, 255}, a(255) = 0+1+1+2+1+2+2+3 = 12.

T. D. Noe, Table of n, a(n) for n = 1..10000
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
E. Pérez Herrero, Psychedelic Geometry Blogspot, CURIOUS SERIES-002

Cf. A001221, A001248, A027750.

a062799 = sum . map a001221 . a027750_row
-- Reinhard Zumkeller, Dec 02 2014

read("transforms") ;
A001221 := proc(n)
        nops(numtheory[factorset](n)) ;
end proc:
omega := [seq(A001221(n),n=1..80)] ;
ones := [seq(1,n=1..80)] ;
DIRICHLET(ones,omega) ; # R. J. Mathar, Sep 23 2011
N:= 1000: # to get a(1) to a(N)
B:= Vector(N,t-> nops(numtheory:-factorset(t))):
A:= Vector(N):
for d from 1 to N do
  md:= d*[$1..floor(N/d)];
  A[md]:= map(`+`,A[md],B[d])
od:
convert(A,list); # Robert Israel, Oct 21 2015

f[n_] := Block[{d = Divisors[n], c = l = 0, k = 2}, l = Length[d]; While[k < l + 1, c = c + Length[ FactorInteger[ d[[k]] ]]; k++ ]; Return[c]]; Table[f[n], {n, 1, 100} ]
omega[n_] := Length[FactorInteger[n]]; SetAttributes[omega, Listable]; omega[1] := 0; A062799[n_] := Plus @@ omega[Divisors[n]] (* Enrique Pérez Herrero, Sep 08 2009 *)

a(n)=my(f=factor(n)[,2],s);forvec(v=vector(#f,i,[0,f[i]]),s+=sum(i=1,#f,v[i]>0));s \\ Charles R Greathouse IV, Oct 15 2015

vector(100, n, sumdiv(n, k, omega(k))) \\ Altug Alkan, Oct 15 2015

a(n) = Sum_{d|n} A001221(d), that is, where d runs over divisors of n.
For squarefree s (i.e., s in A005117), a(s) = omega(s)*2^(omega(s)-1), where omega(n) = A001221(n). Also, for n>1, a(n) <= omega(n)*A000005(n) - 1. - Enrique Pérez Herrero, Sep 08 2009
Let n=Product_{i=1..omega(n)} p(i)^e(i). a(n) = d[Product_{i=1..omega(n)} (1 + e(i)*x)]/dx|x=1. In other words, a(n) = Sum_{m>=1} A146289(n,m)*m. - Geoffrey Critzer, Feb 10 2015
a(A000040(n)) = 1; a(A001248(n)) = 2; a(A030078(n)) = 3; a(A030514(n)) = 4; a(A050997(n)) = 5. - Altug Alkan, Oct 17 2015
a(n) = Sum_{prime p|n} A000005(n/p). - Max Alekseyev, Aug 11 2016
G.f.: Sum_{k>=1} omega(k)*x^k/(1 - x^k), where omega(k) is the number of distinct primes dividing k (A001221). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s)^2*primezeta(s) where primezeta(s) = Sum_{prime p} p^(-s). - Benedict W. J. Irwin, Jul 16 2018

A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 24 2005

T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).
The row length sequence of this table is p(n)=A000041(n) (number of partitions).
In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.
Rows are the same as the rows of A115623, but in reverse order.
From Wolfdieter Lang, Mar 17 2011: (Start)
The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).
See a comment by Giovanni Resta under A000005. (End)
From Gus Wiseman, May 20 2020: (Start)
The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.
Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).
Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.
(End)

			Triangle starts:
  0,
  1,
  1, 1,
  1, 2, 1,
  1, 2, 1, 2, 1,
  1, 2, 2, 2, 2, 2, 1,
  1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.

Robert Price, Table of n, a(n) for n = 0..9295 (first 25 rows).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 10 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A000070.
Row lengths are A000041.
The lengths of these partitions are A036043.
The maxima of these partitions are A049085.
The version for non-reversed partitions is A334440.
The version for colex instead of lex is (also) A334440.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Compositions in Abramowitz-Stegun order are A124734.
Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.

Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

a(n) = A001221(A185974(n)). - Gus Wiseman, May 20 2020

Edited by Franklin T. Adams-Watters, May 29 2006

A166133 After initial 1,2,4, a(n+1) is the smallest divisor of a(n)^2-1 that has not yet appeared in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 21, 20, 19, 18, 17, 24, 23, 22, 69, 28, 27, 26, 25, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 201, 80, 79

Offset: 1

Franklin T. Adams-Watters, Oct 07 2009

The initial 1,2,4 provides the smallest example with this rule that is not simply the integers in order, nor (apparently) ends with all divisors of a(n)^2-1 already present.
Apparently the sequence is infinite and includes every positive integer.
Apr 05 2015: John Mason has computed the first ten million terms. See link to zipped file. - N. J. A. Sloane, Apr 06 2015
The sequence contains many runs of incrementing and decrementing values. In the 1200 steps following the 4, there are 136 increments, 706 decrements, and 358 larger steps. What is the limiting distribution for these steps? [Click the "listen" button to appreciate these runs. - N. J. A. Sloane, Apr 03 2015]
After 3, 198, 270, 570, 522, 600, 822, and 882, we have a(n+1) = a(n)^2-1. Does this happen infinitely often? Cf. A256406, A256407.
A256543 gives numbers m such that a(m+1) = a(m)-1 or a(m+1) = a(m)+1. - Reinhard Zumkeller, Apr 01 2015
If this is a permutation, then A255833 is the inverse permutation. - M. F. Hasler, Apr 01 2015
a(A256703(n)+1) = a(A256703(n))^2 - 1. - Reinhard Zumkeller, Apr 08 2015
For n > 3: a(n) = A027750(a(n-1)^2-1, A256751(n)). - Reinhard Zumkeller, Apr 09 2015

			After a(24) = 22, the divisors of 22^2-1 = 483 are 1, 3, 7, 21, 23, 69, 161, and 483; 1, 3, 7, 21, and 23 have already occurred, so a(25) = 69.

Franklin T. Adams-Watters and N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1203 terms from Franklin T. Adams-Watters)
Hans Havermann, Log plot of 450000+ terms [Produced by Mathematica's ListLogPlot command]
Hans Havermann, Over-sized point-joined (over 250000 terms) graph of the sequence [Heavily clipped, which explains the strange appearance. - _N. J. A. Sloane_, Apr 01 2015]
John Mason, Table of n, a(n) for n = 1..711888 [10 megabytes]
John Mason, Table of n, a(n) for n = 1..2000000 [32 megabytes]
John Mason, Ten million terms (zipped file)
John Mason, Java program to generate this sequence, used to generate 10M terms, and some other associated sequences; it requires splitting into single classes for use.
N. J. A. Sloane and others, "Blog" about A166133

Cf. A166134, A000027, A122280, A005563, A256406, A256407, A027750, A005563, A256557, A256559.
For records see A256403, A256404.
Smallest missing numbers: A256405, A256408, A256409.
Cf. A256541 (first differences), A256543.
Inverse (conjectured): A255833.
Cf. A256564 (smallest prime factors), A244080 (largest prime factors), A256578 (largest proper divisors), A256542 (number of divisors).
Upper envelope: the sequence of pairs (A256422(n),A256423(n)).
Cf. A256703.
Cf. A256751.

import Data.List (delete); import Data.List.Ordered (isect)
a166133 n = a166133_list !! (n-1)
a166133_list = 1 : 2 : 4 : f (3:[5..]) 4 where
   f zs x = y : f (delete y zs) y where
            y = head $ isect (a027750_row' (x ^ 2 - 1)) zs
-- Reinhard Zumkeller, Apr 01 2015

s = {1, 2, 4}; e = 4; Do[d = Divisors[e^2 - 1]; i = 1;
While[MemberQ[s, d[[i]]], i++]; e = d[[i]]; AppendTo[s, e], {19997}]; s (* Hans Havermann, Apr 03 2015 *)

al(n,m=4,u=6)={local(ds,db);
u=bitor(u,1<

A210959 Triangle read by rows in which row n lists the divisors of n starting with 1, n, the second smallest divisor of n, the second largest divisor of n, the third smallest divisor of n, the third largest divisor of n, and so on.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 29 2012

A two-dimensional arrangement of squares has the property that the number of vertices in row n equals the number of divisors of n. So T(n,k) is represented in the structure as the k-th vertex of row n (see the illustration of initial terms).

			Written as an irregular triangle the sequence begins:
1;
1, 2;
1, 3;
1, 4, 2;
1, 5;
1, 6, 2, 3;
1, 7;
1, 8, 2, 4;
1, 9, 3;
1, 10, 2, 5;
1, 11;
1, 12, 2, 6, 3, 4;

Michel Marcus, Rows 1..1000, flattened
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to sigma(n)

Row n has length A000005(n). Row sums give A000203. Right border gives A033677.

Cf. A027750, A161901, A212119.

row(n) = my(d=divisors(n)); vector(#d, k, if (k % 2, d[(k+1)/2], d[#d-k/2+1])); \\ Michel Marcus, Jun 20 2019

A340035 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the divisors of m, with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2020

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2
  1, 3;
  -----
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 3,
  1, 2, 4;
  --------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 3,
  1, 3,
  1, 2, 4,
  1, 5;
--------
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
|   | A027750 |     |       |         |           |  1          |
| D | A027750 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 |     |       |         |  1        |  1 2        |
| I | A027750 |     |       |         |  1        |  1 2        |
| S | A027750 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 |     |       |  1      |  1 2      |  1   3      |
| S | A027750 |     |       |  1      |  1 2      |  1   3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A340032 but here, in the upper zone, every row is A027750 instead of A127093.
Also the above table is the table of A338156 upside down.
The connection with the tower described in A221529 is as follows (n = 7):
|--------|------------------------|
| Level  |                        |
| in the | 7th slice of divisors  |
| tower  |                        |
|--------|------------------------|
|  11    |   1,                   |
|  10    |   1,                   |
|   9    |   1,                   |
|   8    |   1,                   |
|   7    |   1,                   |
|   6    |   1,                   |
|   5    |   1,                   |
|   4    |   1,                   |
|   3    |   1,                   |
|   2    |   1,                   |
|   1    |   1,                   |
|--------|------------------------|
|   7    |   1, 2,                |
|   6    |   1, 2,                |
|   5    |   1, 2,                |
|   4    |   1, 2,                |
|   3    |   1, 2,                |
|   2    |   1, 2,                |
|   1    |   1, 2,                |
|--------|------------------------|
|   5    |   1,    3,             |
|   4    |   1,    3,             |
|   3    |   1,    3,             |
|   2    |   1,    3,             |      Level
|   1    |   1,    3,             |             _
|--------|------------------------|       11   | |
|   3    |   1, 2,    4,          |       10   | |
|   2    |   1, 2,    4,          |        9   | |
|   1    |   1, 2,    4,          |        8   |_|_
|--------|------------------------|        7   |   |
|   2    |   1,          5,       |        6   |_ _|_
|   1    |   1,          5,       |        5   |   | |
|--------|------------------------|        4   |_ _|_|_
|   1    |   1, 2, 3,       6,    |        3   |_ _ _| |_
|--------|------------------------|        2   |_ _ _|_ _|_ _
|   1    |   1,                7; |        1   |_ _ _ _|_|_ _|
|--------|------------------------|
             Figure 1.                            Figure 2.
                                                Lateral view
                                                of the tower.
.
                                                _ _ _ _ _ _ _
                                               |_| | | | |   |
                                               |_ _|_| | |   |
                                               |_ _|  _|_|   |
                                               |_ _ _|    _ _|
                                               |_ _ _|  _|
                                               |       |
                                               |_ _ _ _|
.
                                                  Figure 3.
                                                  Top view
                                                of the tower.
.
Figure 1 shows the terms of the 7th row of the triangle arranged as the 7th slice of the tetrahedron. The left hand column (see figure 1) gives the level of the sum of the divisors in the tower (see figures 2 and 3).

Paolo Xausa, Table of n, a(n) for n = 1..17815 (rows 1..20 of triangle, flattened)

Nonzero terms of A340032.
Row lengths give A006128, n >= 1.
Row sums give A066186, n >= 1.
Cf. A000041, A002260, A027750, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A206437, A207031, A207383, A221529, A221530, A221531, A221649, A236104, A237593, A245092, A245095, A221649, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.

A340035row[n_]:=Flatten[Array[ConstantArray[Divisors[#],PartitionsP[n-#]]&,n]];
nrows=7;Array[A340035row,nrows] (* Paolo Xausa, Jun 20 2022 *)

A119347 Number of distinct sums of distinct divisors of n. Here 0 (as the sum of an empty subset) is excluded from the count.

Original entry on oeis.org

1, 3, 3, 7, 3, 12, 3, 15, 7, 15, 3, 28, 3, 15, 15, 31, 3, 39, 3, 42, 15, 15, 3, 60, 7, 15, 15, 56, 3, 72, 3, 63, 15, 15, 15, 91, 3, 15, 15, 90, 3, 96, 3, 63, 55, 15, 3, 124, 7, 63, 15, 63, 3, 120, 15, 120, 15, 15, 3, 168, 3, 15, 59, 127, 15, 144, 3, 63, 15, 142, 3, 195, 3, 15, 63, 63

Offset: 1

Emeric Deutsch, May 15 2006

nonn

If a(n)=sigma(n) (=sum of the divisors of n =A000203(n); i.e. all numbers from 1 to sigma(n) are sums of distinct divisors of n), then n is called a practical number (A005153). The actual sums obtained from the divisors of n are given in row n of the triangle A119348.
The records appear to occur at the highly abundant numbers, A002093, excluding 3 and 10. For n in A174533, a(n) = sigma(n)-2. - T. D. Noe, Mar 29 2010
The indices of records occur at the highly abundant numbers, excluding 3 and 10, if Jaycob Coleman's conjecture at A002093 that all these numbers are practical numbers (A005153) is true. - Amiram Eldar, Jun 13 2020
Zumkeller numbers A083207 give the positions of even terms in this sequence (likewise, the positions of odd terms in A308605). - Antti Karttunen and Ilya Gutkovskiy, Nov 29 2024

			a(5)=3 because the divisors of 5 are 1 and 5 and all the possible sums: are 1,5 and 6; a(6)=12 because we can form all sums 1,2,...,12 by adding up the terms of a nonempty subset of the divisors 1,2,3,6 of 6.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Jon Maiga, Computer-generated formulas for A119347, Sequence Machine.
B. M. Stewart, Sums of distinct divisors, American Journal of Mathematics 76 (1954), pp. 779-785.

One less than A308605.
Cf. A000203, A002093, A005153, A027750, A030057, A033630, A093890, A100587, A119348, A193279, A225561, A237290, A378447, A378450, A378451.
Cf. A083207 (positions of even terms).

import Data.List (subsequences, nub)
a119347 = length . nub . map sum . tail . subsequences . a027750_row'
-- Reinhard Zumkeller, Jun 27 2015

with(numtheory): with(linalg): a:=proc(n) local dl,t: dl:=convert(divisors(n),list): t:=tau(n): nops({seq(innerprod(dl,convert(2^t+i,base,2)[1..t]),i=1..2^t-1)}) end: seq(a(n),n=1..90);

a[n_] := Total /@ Rest[Subsets[Divisors[n]]] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Jan 27 2018 *)

A119347(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); sum(i=1,poldegree(p),(0Antti Karttunen, Nov 28 2024

A119347(n) = { my(c=[0]); fordiv(n, d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A119347(n):
    c = {0}
    for d in divisors(n,generator=True):
        c |=  {a+d for a in c}
    return len(c)-1 # Chai Wah Wu, Jul 05 2023

For n > 1, 3 <= a(n) <= sigma(n). - Charles R Greathouse IV, Feb 11 2019
For p prime, a(p) = 3. For k >= 0, a(2^k) = 2^(k + 1) - 1. - Ctibor O. Zizka, Oct 19 2023
From Antti Karttunen, Nov 29 2024: (Start)
a(n) = A308605(n)-1.
a(n) = 2*(A237290(n)/A000203(n)) - 1. [Found by Sequence Machine. See A237290.]
a(n) <= A100587(n).
(End)

Definition clarified by Antti Karttunen, Nov 29 2024

A007948 Largest cubefree number dividing n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 9, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 18, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 36, 73

Offset: 1

R. Muller

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative functions.
Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Cf. A003557, A004709, A007947, A027748, A058035, A062378, A065465, A124010, A197863.

a007948 = last . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012, Jan 06 2012

Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 73}] (* Michael De Vlieger, Jul 18 2017 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = min(f[i, 2], 2)); factorback(f); \\ Michel Marcus, Jun 09 2014
(Scheme, with memoization-macro definec) (definec (A007948 n) (if (= 1 n) n (* (expt (A020639 n) (min 2 (A067029 n))) (A007948 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Multiplicative with a(p^e) = p^(min(e, 2)). - David W. Wilson, Aug 01 2001
a(n) = max{A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)}. - Reinhard Zumkeller, May 27 2012
a(n) = A071773(n)*A007947(n). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
a(n) = n / A062378(n) = n / A003557(A003557(n)). - Antti Karttunen, Nov 28 2017
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Oct 13 2022

More terms from Henry Bottomley, Jun 18 2001

cp's OEIS Frontend

A067513 Number of divisors d of n such that d+1 is prime.

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A187202 The bottom entry in the difference table of the divisors of n.

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A336812 Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.

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A062799 Inverse Möbius transform of the numbers of distinct prime factors (A001221).

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A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

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A166133 After initial 1,2,4, a(n+1) is the smallest divisor of a(n)^2-1 that has not yet appeared in the sequence.

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