A122934 -id:A122934 - cp's OEIS frontend

A032741 a(0) = 0; for n > 0, a(n) = number of proper divisors of n (divisors of n which are less than n).

0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 7, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 7, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 11, 1, 3, 5, 6, 3, 7, 1, 5, 3, 7, 1, 11, 1, 3, 5, 5, 3, 7, 1, 9, 4, 3, 1, 11, 3, 3, 3, 7, 1, 11, 3, 5, 3, 3, 3, 11, 1, 5, 5

Offset: 0

Patrick De Geest, May 15 1998

Number of d < n which divide n.
Call an integer k between 1 and n a "semi-divisor" of n if n leaves a remainder of 1 when divided by k, i.e., n == 1 (mod k). a(n) gives the number of semi-divisors of n+1. - Joseph L. Pe, Sep 11 2002
a(n+1) is also the number of k, 0 <= k <= n-1, such that C(n,k) divides C(n,k+1). - Benoit Cloitre, Oct 17 2002
a(n+1) is also the number of factors of the n-th degree polynomial x^n + x^(n-1) + x^(n-2) + ... + x^2 + x + 1. Example: 1 + x + x^2 + x^3 = (1+x)(1+x^2) implies a(4)=2.
a(n) is also the number of factors of the n-th Fibonacci polynomial. - T. D. Noe, Mar 09 2006
Number of partitions of n into 2 parts with the second dividing the first. - Franklin T. Adams-Watters, Sep 20 2006
Number of partitions of n+1 into exactly one q and at least one q+1. Example: a(12)=5; indeed, we have 13 = 7 + 6 = 5 + 4 + 4 = 4 + 3 + 3 + 3 = 3 + 2 + 2 + 2 + 2 + 2 = 2 + 11*1.
Differences of A002541. - George Beck, Feb 12 2012
For n > 1: number of ones in row n+1 of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
For n > 0, a(n) is the number of strong divisors of n. - Omar E. Pol, May 03 2015
a(n) is also the number of factors of the (n-1)-th degree polynomial ((x+1)^n-1)/x. Example: for n=6, ((x+1)^6-1)/x = x^5 + 6*x^4 + 15*x^3 + 20*x^2 + 15*x + 6 = (2+x)(1+x+x^2)(3+3x+x^2) implies a(6)=3. - Federico Provvedi, Oct 09 2018
Consider the polynomial P(n,z) = Sum_{i=1..q} d(i)*z^(i-1) where d(1), d(2), ..., d(q) are are the q ordered divisors of n. The sequence lists the numbers of zeros of P(n,z) strictly inside the unit circle. - Michel Lagneau, Apr 06 2025

			a(6) = 3 since the proper divisors of 6 are 1, 2, 3.

André Weil, Number Theory, An approach through history, From Hammurapi to Legendre, Birkhäuser, 1984, page 5.

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Proper divisors.

Column 2 of A122934.
Cf. A003238, A001065, A027749, A027751 (list of proper divisors).
Cf. A001620, A039653, A051778, A070824, A091818.

Concatenation([0],List([1..100],n->Tau(n)-1)); # Muniru A Asiru, Oct 09 2018

a032741 n = if n == 0 then 0 else a000005 n - 1
-- Reinhard Zumkeller, Jul 31 2014

A032741 := proc(n)
    if n = 0 then
        0 ;
    else
        numtheory[tau](n)-1 ;
    end if;
end proc: # R. J. Mathar, Feb 03 2013

Prepend[DivisorSigma[0, Range[99]]-1, 0] (* Jayanta Basu, May 25 2013 *)

PARI
```
a(n) = if(n<1,0,numdiv(n)-1)
```

{a(n)=polcoeff(2*sum(m=1,n\2+1,sumdiv(m,d,log(1-x^(m/d) +x*O(x^n) )^(2*d)/(2*d)!)), n)} \\ Paul D. Hanna, Aug 21 2014

from sympy import divisor_count
def A032741(n): return divisor_count(n)-1 if n else 0 # Chai Wah Wu, Mar 14 2023

a(n) = tau(n)-1 = A000005(n)-1. Cf. A039653.
G.f.: Sum_{n>=1} x^(2*n)/(1-x^n). - Michael Somos, Apr 29 2003
G.f.: Sum_{i>=1} (1-x^i+x^(2*i))/(1-x^i). - Jon Perry, Jul 03 2004
a(n) = Sum_{k=1..floor(n/2)} A051731(n-k,k). - Reinhard Zumkeller, Nov 01 2009
G.f.: 2*Sum_{n>=1} Sum_{d|n} log(1 - x^(n/d))^(2*d) / (2*d)!. - Paul D. Hanna, Aug 21 2014
Dirichlet g.f.: zeta(s)*(zeta(s)-1). - Geoffrey Critzer, Dec 06 2014
a(n) = Sum_{k=1..n-1} binomial((n-1) mod k, k-1). - Wesley Ivan Hurt, Sep 26 2016
a(n) = Sum_{i=1..n-1} floor(n/i)-floor((n-1)/i). - Wesley Ivan Hurt, Nov 15 2017
a(n) = Sum_{i=1..n-1} 1-sign(i mod (n-i)). - Wesley Ivan Hurt, Sep 27 2018
Sum_{k=1..n} a(k) ~ n*log(n) + 2*(gamma - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022

Typos in definition corrected by Omar E. Pol, Dec 13 2008

A003238 Number of rooted trees with n vertices in which vertices at the same level have the same degree.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 16, 19, 26, 27, 40, 41, 53, 61, 77, 78, 104, 105, 134, 147, 175, 176, 227, 233, 275, 294, 350, 351, 438, 439, 516, 545, 624, 640, 774, 775, 881, 924, 1069, 1070, 1265, 1266, 1444, 1521, 1698, 1699

Offset: 1

N. J. A. Sloane

Also, number of sequences of positive integers b_1, b_2, ..., b_k such that 1 + b_1*(1 + b_2*(...(1 + b_k) ... )) = n. If you take mu(b_1)*mu(b_2)*...*mu(b_k) for each sequence you get 1's 0's and -1's. Add them up and you get the terms for A007554. - Christian G. Bower, Oct 15 1998
Note that this applies also to planar rooted trees and other similar objects (mountain ranges, parenthesizations) encoded by A014486. - Antti Karttunen, Sep 07 2000
Equals sum of (n-1)-th row terms of triangle A152434. - Gary W. Adamson, Dec 04 2008
Equals the eigensequence of A051731, the inverse binomial transform. - Gary W. Adamson, Dec 26 2008
From Emeric Deutsch, Aug 18 2012: (Start)
The considered rooted trees are called generalized Bethe trees; in the Goldberg-Livshitz reference they are called uniform trees.
Also, a(n) = number of partitions of n-1 in which each part is divisible by the next. Example: a(5)=5 because we have 4, 31, 22, 211, and 1111.
There is a simple bijection between generalized Bethe trees with n+1 vertices and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges --- sum of parts; root degree --- last part; number of leaves --- first part; height --- number of parts. (End)
a(n+1) = a(n) + 1 if and only if n is prime. - Jon Perry, Nov 24 2012
According to the MathOverflow link, log(a(n)) ~ log(4)*log(n)^2, and a more precise asymptotic expansion is similar to that of A018819 and hence A000123, so the conjecture in the Formula section is partly correct. - Andrey Zabolotskiy, Jan 22 2017

			a(4) = 3 because we have the path P(4), the tree Y, and the star \|/ . - _Emeric Deutsch_, Aug 18 2012
The planted achiral trees with up to 7 nodes are:
 1  -
 1  (-)
 2  (--),     ((-))
 3  (---),    ((--)),      (((-)))
 5  (----),   ((-)(-)),    ((---)),    (((--))),     ((((-))))
 6  (-----),  ((----)),    (((-)(-))), (((---))),    ((((--)))), (((((-)))))
10 (------), ((-)(-)(-)), ((--)(--)), (((-))((-))), ((-----)),  (((----))), ((((-)(-)))), ((((---)))), (((((--))))), ((((((-)))))). - _Gus Wiseman_, Jan 12 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 1..10000
G. Gati, F. Harary, and R. W. Robinson, Line colored trees with extendable automorphisms, Acta Mathematica Scientia 2.1 (1982), 105-110. (Annotated scanned copy)
M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
B. S. Kochkarev, Absolutely symmetric trees and complexity of natural number, arXiv:1205.0344 [math.CO], 2012.
MathOverflow, Are the asymptotics of A003238 known?
O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms
Gus Wiseman, Planted achiral trees n=1..10.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A007439, A007554, A057546, A152434, A051731, A002033, A027750, A281487, A000123.
Row sums of A122934 (offset by 1).
Cf. A004111, A280994.

a003238 n = a003238_list !! (n-1)
a003238_list = 1 : f 1 where
   f x = (sum (map a003238 $ a027750_row x)) : f (x + 1)
-- Reinhard Zumkeller, Dec 20 2014

a = new Array();
for (i = 1; i < 50; i++) a[i] = 1;
for (i = 3; i < 50; i++) for (j = 2; j < i; j++) if (i % j == 1) a[i] += a[j];
document.write(a + "
"); // Jon Perry, Nov 20 2012

with(numtheory): aa := proc (n) if n = 0 then 1 else add(aa(divisors(n)[i]-1), i = 1 .. tau(n)) end if end proc: a := proc (n) options operator, arrow: aa(n-1) end proc: seq(a(n), n = 1 .. 48); # Emeric Deutsch, Aug 18 2012
A003238:= proc(n) option remember; uses numtheory; add(A003238(m),m=divisors(n-1)) end proc;
A003238(1):= 1;
[seq(A003238(n),n=1..48)]; # Robert Israel, Mar 10 2014

(* b = A068336 *) b[1] = 1; b[n_] := b[n] = 1 + Sum[b[k], {k, Divisors[n-1]}]; a[n_] := b[n]/2; a[1] = 1; Table[ a[n], {n, 1, 48}] (* Jean-François Alcover, Dec 20 2011, after Ralf Stephan *)
achi[n_]:=If[n===1,1,Total[achi/@Divisors[n-1]]];Array[achi,50] (* Gus Wiseman, Jan 12 2017 *)

seq(n) = {my(v=vector(n)); v[1]=1; for(i=2, n, v[i]=sumdiv(i-1, d, v[d])); v} \\ Andrew Howroyd, Jun 08 2025

Shifts one place left under inverse Moebius transform: a(n+1) = Sum_{k|n} a(k).
Conjecture: log(a(n)) is asymptotic to c*log(n)^2 where 0.4 < c < 0.5 - Benoit Cloitre, Apr 13 2004
For n > 1, a(n) = (1/2) * A068336(n) and Sum_{k = 1..n} a(k) = A003318(n). - Ralf Stephan, Mar 27 2004
Generating function P(x) for the sequence with offset 2 obeys P(x) = x^2*(1 + Sum_{n >= 1} P(x^n)/x^n). [Harary & Robinson]. - R. J. Mathar, Sep 28 2011
a(n) = 1 + sum of a(i) such that n == 1 (mod i). - Jon Perry, Nov 20 2012
From Ilya Gutkovskiy, Apr 28 2019: (Start)
G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)).
L.g.f.: -log(Product_{n>=1} (1 - x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n. (End)

Description improved by Christian G. Bower, Oct 15 1998

A122651 Number of partitions of n into distinct parts, with each part divisible by the next.

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Franklin T. Adams-Watters and Vladeta Jovovic, Sep 21 2006

			a(9)  = 4 : [9], [8,1], [6,3], [6,2,1].
a(15) = 6 : [15], [14,1], [12,3], [12,2,1], [10,5], [8,4,2,1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003238, A122934, A167439, A167865, A167866, A184999.

A122651r := proc(n,pmax,dv) option remember ; local a,d ; a := 0 ; for d in dv do if d = n and d <= pmax then a := a+1 ; elif d < pmax and n-d > 0 then a := a+A122651r(n-d,d-1,numtheory[divisors](d) minus {d} ) ; fi; od: a ; end: A122651 := proc(n) local i; A122651r(n,n, convert([seq(i,i=1..n)],set) ) ; end: for n from 1 to 120 do printf("%d,",A122651(n)) ; od:  # R. J. Mathar, May 22 2009
# second Maple program:
with(numtheory):
b:= proc(n) option remember;
      `if`(n=0, 1, add(b((n-d)/d), d=divisors(n) minus{1}))
    end:
a:= n-> `if`(n=0, 1, b(n)+b(n-1));
seq(a(n), n=0..200);  # Alois P. Heinz, Mar 28 2011

b[0] = 1; b[n_] := b[n] = Sum[b[(n - d)/d], {d, Divisors[n] // Rest}]; a[0] = 1; a[n_] := b[n] + b[n-1]; Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Mar 26 2013, after Alois P. Heinz *)

{ a(n,m=0) = local(r=0); if(n==0,return(1)); fordiv(n,d, if(d<=m,next); r+=a((n-d)\d,1); ); r } /* Max Alekseyev */

For n>0, a(n) = A167865(n) + A167865(n-1).

More terms from R. J. Mathar, May 22 2009

a(0)=1 prepended by Max Alekseyev, Nov 13 2009

A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

Original entry on oeis.org

1, 2, 3, 1, 4, 5, 2, 1, 6, 7, 3, 1, 8, 2, 9, 4, 1, 10, 11, 5, 3, 2, 1, 12, 13, 6, 1, 14, 4, 2, 15, 7, 3, 1, 16, 17, 8, 5, 2, 1, 18, 19, 9, 4, 3, 1, 20, 6, 2, 21, 10, 1, 22, 23, 11, 7, 5, 3, 2, 1, 24, 4, 25, 12, 1, 26, 8, 2, 27, 13, 6, 3, 1, 28, 29, 14, 9, 5, 4, 2, 1

Offset: 2

Gus Wiseman, Nov 03 2022

			Grouping by sum gives:
   2:  1
   3:  2
   4:  3 1
   5:  4
   6:  5 2 1
   7:  6
   8:  7 3 1
   9:  8 2
  10:  9 4 1
  11: 10
  12: 11 5 3 2 1
  13: 12
  14: 13 6 1
  15: 14 4 2
  16: 15 7 3 1
  17: 16
  18: 17 8 5 2 1

Row-lengths are A032741.
This is A208460/A027751.
A ranking of divisible pairs is A318990, proper A339005.
A different ordering is A358103 = A358104 / A358105.
A000041 counts partitions, strict A000009.
A001358 lists semiprimes, squarefree A006881.
A318991 ranks divisor-chains.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000837, A003238, A122934.

Table[Divide@@@Select[IntegerPartitions[n,{2}],Divisible@@#&],{n,2,30}]

a(n) = A208460(n)/A027751(n).

A214575 Triangle read by rows: T(n,k) is the number of partitions of n in which each part is divisible by the next and have first part equal to k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 4, 2, 4, 1, 2, 1, 1, 1, 4, 3, 4, 1, 3, 1, 1, 1, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 1, 1, 6, 4, 9, 2, 7, 1, 4, 2, 2, 1, 1, 1, 6, 4, 9, 2, 7, 1, 4, 2, 2, 1, 1, 1, 1, 7, 4, 12, 2, 9, 2, 6, 2, 3, 1, 2, 1, 1

Offset: 1

Emeric Deutsch, Aug 18 2012

T(n,k) is also the number of generalized Bethe trees with n edges and k leaves.
A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree; they are called uniform trees in the Goldberg and Livshits reference.
There is a simple bijection between generalized Bethe trees with n edges and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges --- sum of parts; root degree --- last part; number of leaves --- first part; height --- number of parts.
Sum of entries in row n is A003238(n+1).
Apparently, Sum(k*T(n,k), k>=1) = A038046(n+1).

			T(7,4)=2 because we have (4,2,1) and (4,1,1,1).
Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 2, 1, 1, 1;
  1, 3, 2, 2, 1, 1;

Seiichi Manyama, Rows n = 1..50, flattened
M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.

An augmented version is A301343.

Cf. A122934, A214576.

with(numtheory): T := proc (n, k) if k = 1 then 1 elif n < k then 0 else add(T(n-k, divisors(k)[j]), j = 1 .. tau(k)) end if end proc: for n to 18 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

T(n,1)=1; T(n,k) = Sum_{j|k}T(n-k,j); T(n,k)=0 if k>n.

A049822 a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 4, 4, 3, 4, 5, 4, 6, 5, 4, 6, 5, 6, 9, 6, 3, 6, 9, 7, 7, 8, 5, 10, 7, 6, 9, 7, 8, 11, 8, 6, 9, 10, 7, 12, 7, 8, 14, 8, 3, 10, 12, 13, 10, 11, 5, 10, 12, 12, 13, 8, 3, 14, 11, 8, 15, 11, 13, 16, 7, 9, 9, 14, 7, 14, 11, 9, 16, 12, 11, 15, 7, 14, 16, 11, 3, 18, 17, 10, 9, 12

Offset: 1

Clark Kimberling

Number of partitions of n into 3 summands 0 < a <= b <= c with b/a and c/b integers.

a(n) is the number of 1's in the n-th row of array T given by A049816. E.g., there are 5 numbers k from 1 to 13 for which the Euclidean algorithm on (13, k) has exactly 1 nonzero remainder; hence a(13) = 5.

			a(6) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.
a(100) = 20 because there are 20 partitions of 100 in 3 summands 0 < a <= b <= c with integer b/a and c/b: {a, b, c} = {1, 1, 98}, {1, 3, 96}, {1, 9, 90}, {1, 11, 88}, {1, 33, 66}, {2, 2, 96}, {2, 14, 84}, {4, 4, 92}, {4, 8, 88}, {4, 12, 84}, {4, 16, 80}, {4, 24, 72}, {4, 32, 64}, {4, 48, 48}, {5, 5, 90}, {10, 10, 80}, {10, 30, 60}, {20, 20, 60}, {20, 40, 40}, {25, 25, 50}.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Column 3 of A122934.
Cf. A000005, A003238, A057427.
Cf. A069905 (number of partitions of n into 3 positive parts).

a[n_] := 1 - DivisorSigma[0, n] + DivisorSum[n, If[# == 1, 0, DivisorSigma[ 0, # - 1]]& ]; Array[a, 90] (* Jean-François Alcover, Dec 02 2015 *)

a(n) = 1 - numdiv(n) + sumdiv(n, d, if (d==1, 0, numdiv(d-1))); \\ Michel Marcus, Oct 01 2013

Additional comments from Vladeta Jovovic, Aug 23 2003, Zak Seidov, Aug 31 2006 and Franklin T. Adams-Watters, Sep 20 2006

Edited by N. J. A. Sloane, Sep 21 2006

A121895 Number of partitions of n into 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 2, 5, 4, 6, 4, 7, 5, 10, 5, 8, 6, 11, 8, 13, 6, 12, 7, 13, 9, 15, 8, 16, 10, 17, 10, 14, 10, 20, 11, 14, 10, 23, 10, 22, 12, 21, 15, 20, 8, 21, 12, 23, 18, 24, 11, 20, 15, 30, 18, 21, 8, 28, 14, 21, 18, 32, 16, 34, 16, 22, 15, 28, 14, 33, 14, 22, 20, 31, 18, 32, 15

Offset: 1

Zak Seidov, Sep 01 2006

nonn

			a(36)=20 because there are 20 partitions of 36 in 4 summands a>=b>=c>=d>0 with integer a/b, b/c and c/d:
{33, 1, 1, 1}, {32, 2, 1, 1}, {30, 2, 2, 2}, {28, 4, 2, 2}, {27, 3, 3, 3}, {25, 5, 5, 1}, {24, 8, 2, 2}, {24, 6, 3, 3}, {24, 4, 4, 4}, {21, 7, 7, 1}, {20, 10, 5, 1}, {18, 6, 6, 6}, {17, 17, 1, 1}, {16, 16, 2, 2}, {16, 8, 8, 4}, {15, 15, 5, 1}, {15, 15, 3, 3}, {14, 14, 7, 1}, {12, 12, 6, 6}, {9, 9, 9, 9}.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A026810 = number of partitions of n into exactly 4 parts.
Column 4 of A122934.
Cf. A049822, A003238.

a(n) = Sum_{d|n, d>1} A122935(d-1). - Franklin T. Adams-Watters, Sep 20 2006

A214576 Triangle read by rows: T(n,k) is the number of partitions of n in which each part is divisible by the next and have last part equal to k (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 6, 2, 1, 0, 0, 1, 10, 0, 0, 0, 0, 0, 1, 11, 3, 0, 1, 0, 0, 0, 1, 16, 0, 2, 0, 0, 0, 0, 0, 1, 19, 5, 0, 0, 1, 0, 0, 0, 0, 1, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 27, 6, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Emeric Deutsch, Aug 18 2012

T(n,k) is also the number of generalized Bethe trees with n edges and root degree k.
A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree; they are called uniform trees in the Goldberg and Livshits reference.
There is a simple bijection between generalized Bethe trees with n edges and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: root degree --- last part; number of leaves --- first part; height --- number of parts.
Sum of entries in row n is A003238(n+1).

			T(9,3)=2 because we have (6,3) and (3,3,3).
Triangle starts:
   1;
   1, 1;
   2, 0, 1;
   3, 1, 0, 1;
   5, 0, 0, 0, 1;
   6, 2, 1, 0, 0, 1;
  10, 0, 0, 0, 0, 0, 1;

Seiichi Manyama, Rows n = 1..140, flattened
M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.

Cf. A003238, A122934, A214575.

with(numtheory): a := proc (n) if n = 0 then 1 else add(a(divisors(n)[j]-1), j = 1 .. tau(n)) end if end proc: T := proc (n, k) if type(n/k, integer) = true then a(n/k-1) else 0 end if end proc: for n to 18 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form

T(n,k)=a(n/k -1) if k|n and = 0 otherwise; here a(n) is defined by a(0)=1, a(n) = sum_{j|n}a(j-1). We have a(n) = A003238(n+1) = number of partitions of n in which each part is divisible by the next one.

cp's OEIS Frontend

A032741 a(0) = 0; for n > 0, a(n) = number of proper divisors of n (divisors of n which are less than n).

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A003238 Number of rooted trees with n vertices in which vertices at the same level have the same degree.

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A122651 Number of partitions of n into distinct parts, with each part divisible by the next.

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A358106 Quotient of the n-th divisible pair, where pairs are ordered first by sum and then by denominator.

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A214575 Triangle read by rows: T(n,k) is the number of partitions of n in which each part is divisible by the next and have first part equal to k (1 <= k <= n).

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A049822 a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1).

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