A168020 -id:A168020 - cp's OEIS frontend

A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 7, 0, 0, 0, 1, 11, 3, 2, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 22, 5, 0, 2, 0, 0, 0, 1, 30, 0, 3, 0, 0, 0, 0, 0, 1, 42, 7, 0, 0, 2, 0, 0, 0, 0, 1, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 77, 11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Nov 20 2009, Nov 21 2009

The row-reversed version is A168016.

Also see A168020.

			Triangle begins:
==============================================
...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
==============================================
n=1 ..... 1,
n=2 ..... 2, 1,
n=3 ..... 3, 0, 1,
n=4 ..... 5, 2, 0, 1,
n=5 ..... 7, 0, 0, 0, 1,
n=6 .... 11, 3, 2, 0, 0, 1,
n=7 .... 15, 0, 0, 0, 0, 0, 1,
n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).

Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.

T[n_, k_]:= If[IntegerQ[n/k], PartitionsP[n/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def A168021(n,k): return number_of_partitions(n/k) if (n%k)==0 else 0
flatten([[A168021(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.
Sum_{k=1..n} T(n, k) = A047968(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = 2*A000012(n).
T(2*n-1, n+1) = A000007(n-2). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 0, 2, 5, 1, 0, 0, 0, 7, 1, 0, 0, 2, 3, 11, 1, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 2, 0, 5, 22, 1, 0, 0, 0, 0, 0, 3, 0, 30, 1, 0, 0, 0, 0, 2, 0, 0, 7, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101

Offset: 1

Omar E. Pol, Nov 21 2009

			Triangle begins:
==============================================
.... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
==============================================
n=1 ....................................... 1,
n=2 ................................... 1,  2,
n=3 ............................... 1,  0,  3,
n=4 ............................ 1, 0,  2,  5,
n=5 ......................... 1, 0, 0,  0,  7,
n=6 ...................... 1, 0, 0, 2,  3, 11,
n=7 ................... 1, 0, 0, 0, 0,  0, 15,
n=8 ................ 1, 0, 0, 0, 2, 0,  5, 22,
n=9 ............. 1, 0, 0, 0, 0, 0, 3,  0, 30,
n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0,  7, 42,
n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0,  0, 56,
n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000005, A000007, A000041, A035363, A035444, A047968, A135010.

Cf. A138121, A168014, A168015, A168020, A168021, A168111.

T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def T(n,k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0
flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

T(n, k) = A000041(n/k) if k|n; otherwise T(n,k) = 0.
T(n, n) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = A000007(n-1).
Sum_{k=1..n} T(n, k) = A047968(n).
Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)

Edited and extended by Max Alekseyev, May 07 2010

A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 5, 1, 7, 1, 2, 3, 11, 1, 15, 1, 2, 5, 22, 1, 3, 30, 1, 2, 7, 42, 1, 56, 1, 2, 3, 5, 11, 77, 1, 101, 1, 2, 15, 135, 1, 3, 7, 176, 1, 2, 5, 22, 231, 1, 297, 1, 2, 3, 11, 30, 385, 1, 490, 1, 2, 5, 7, 42, 627, 1, 3, 15, 792, 1, 2, 56, 1002

Offset: 1

Omar E. Pol, Nov 22 2009

Positive values of triangle A168016.
The number of terms of row n is equal to the number of divisors of n: A000005(n).
Note that the last term of each row is the number of partitions of n: A000041(n).
Also, it appears that row n lists the partition numbers of the divisors of n. [Omar E. Pol, Nov 23 2009]

			Consider row n=8: (1, 2, 5, 22). The divisors of 8 listed in decreasing order are 8, 4, 2, 1 (see A056538). There is 1 partition of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8), (4+4)}; 5 partitions of 8 into parts divisible by 2: {(8), (6+2), (4+4), (4+2+2), (2+2+2+2)}; and 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then row 8 is formed by 1, 2, 5, 22.
Triangle begins:
1;
1,  2;
1,  3;
1,  2,  5;
1,  7;
1,  2,  3, 11;
1, 15;
1,  2,  5, 22;
1,  3, 30;
1,  2,  7, 42;
1, 56;
1,  2,  3,  5, 11, 77;

Alois P. Heinz, Rows n = 1..1400, flattened
Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Row sums give A047968.

Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.

with(numtheory):
b:= proc(n, i, d) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-d, d) +b(n-i, i, d)
      fi
    end:
T:= proc(n) local l;
      l:= sort([divisors(n)[]],`>`);
      seq(b(n, n, l[i]), i=1..nops(l))
    end:
seq(T(n), n=1..30); # Alois P. Heinz, Oct 21 2011

b[n_, i_, d_] := b[n, i, d] = Which[n<0, 0, n==0, 1, i<1, 0, True, b[n, i - d, d] + b[n-i, i, d]]; T[n_] := Module[{l = Divisors[n] // Reverse}, Table[b[n, n, l[[i]]], {i, 1, Length[l]}]]; Table[T[n], {n, 1, 30}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *)

A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 2, 1, 7, 1, 11, 3, 2, 1, 15, 1, 22, 5, 2, 1, 30, 3, 1, 42, 7, 2, 1, 56, 1, 77, 11, 5, 3, 2, 1, 101, 1, 135, 15, 2, 1, 176, 7, 3, 1, 231, 22, 5, 2, 1, 297, 1, 385, 30, 11, 3, 2, 1, 490, 1, 627, 42, 7, 5, 2, 1, 792, 15, 3, 1, 1002, 56, 2, 1, 1255, 1, 1575, 77, 22, 11, 5, 3, 2

Offset: 1

Omar E. Pol, Nov 22 2009

Positive values of triangle A168021.
Note that column 1 lists the numbers of partitions A000041(n).
Row n has A000005(n) terms.
Also, it appears that row n lists the partition numbers of the divisors of n, in decreasing order. [Omar E. Pol, Nov 23 2009]

			For example:
Consider row 8: (22, 5, 2, 1). The divisors of 8 are 1, 2, 4, 8 (see A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22); 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}; 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}; and 1 partition of 8 into parts divisible by 8. Then row 8 is formed by 22, 5, 2, 1.
Triangle begins:
1;
2, 1;
3, 1;
5, 2, 1;
7, 1;
11, 3, 2, 1;
15, 1;
22, 5, 2, 1;
30, 3, 1;
42, 7, 2, 1;
56, 1;
77, 11, 5, 3, 2, 1;

Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Row sums give A047968.

Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.

A168018 := proc(n) local dvs,p,i,d,a,pp,divs,par; dvs := sort(convert(numtheory[divisors](n),list)) ; p := combinat[partition](n) ; for i from 1 to nops(dvs) do d := op(i,dvs) ; a := 0 ; for pp in p do divs := true; for par in pp do if par mod d <> 0 then divs := false; end if; end do ; if divs then a := a+1 ; end if; end do ; printf("%d,",a) ; end do ; end proc: for n from 1 to 40 do A168018(n) ; end do : # R. J. Mathar, Feb 05 2010

Terms beyond row 12 from R. J. Mathar, Feb 05 2010

A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 0, 0, 0, 1, 7, 2, 1, 0, 0, 1, 11, 0, 0, 0, 0, 0, 1, 15, 3, 0, 1, 0, 0, 0, 1, 22, 0, 2, 0, 0, 0, 0, 0, 1, 30, 5, 0, 0, 1, 0, 0, 0, 0, 1, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Omar E. Pol, Nov 21 2009

Note that column k lists each partition number A000041 followed by k zeros. See also A168020 and A168021.

Let A(n,k) denote the number of partitions of n into parts divisible by k+1. Let p(n) denote the number of partitions of n. If k+1 is a divisor of n then A(n,k) = p(n/(k+1)) otherwise A(n,k) = 0. [Conjectured by Omar E. Pol, Nov 25 2009] - this is trivial, just divide each part size by k - Franklin T. Adams-Watters, May 14 2010.

			The array, A(n, k), begins:
==================================================
... Column k: 0.. 1. 2. 3. 4. 5. 6. 7. 8. 9 10 11
. Row ...........................................
...n ............................................
==================================================
.. 0 ........ 1,  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
.. 1 ........ 1,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 2 ........ 2,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 3 ........ 3,  0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
.. 4 ........ 5,  2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,
.. 5 ........ 7,  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0,
.. 6 ....... 11,  3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0,
.. 7 ....... 15,  0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
.. 8 ....... 22,  5, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0,
.. 9 ....... 30,  0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0,
. 10 ....... 42,  7, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0,
. 11 ....... 56,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,
. 12 ....... 77, 11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...
Antidiagonal triangle, T(n, k), begins as:
   1;
   1, 1;
   2, 0, 1;
   3, 1, 0, 1;
   5, 0, 0, 0, 1;
   7, 2, 1, 0, 0, 1;
  11, 0, 0, 0, 0, 0, 1;
  15, 3, 0, 1, 0, 0, 0, 1;
  22, 0, 2, 0, 0, 0, 0, 0, 1;
  30, 5, 0, 0, 1, 0, 0, 0, 0, 1;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A000041, A035363, A083710, A135010.

Cf. A168016, A168120, A168021, A168121.

T[n_, k_]:= If[IntegerQ[(n-k)/(k+1)], PartitionsP[(n-k)/(k+1)], 0];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 13 2023 *)

def A168019(n,k): return number_of_partitions((n-k)/(k+1)) if ((n-k)%(k+1))==0 else 0
flatten([[A168019(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 13 2023

From G. C. Greubel, Jan 13 2023: (Start)
A(n, k) = A000041(n/(k+1)) if (k+1)|n, otherwise 0 (array).
T(n, k) = A000041((n-k)/(k+1)) if (k+1)|(n-k), otherwise 0 (antidiagonals).
A(n, 0) = T(n, 0) = A000041(n).
T(2*n, n) = A(n, n) = A000007(n).
Sum_{k=0..n} T(n, k) = A083710(n+1). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

Edited by Franklin T. Adams-Watters, May 14 2010

A168111 Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 8, 4, 10, 1, 22, 1, 18, 11, 30, 1, 47, 1, 57, 19, 59, 1, 121, 8, 104, 34, 158, 1, 242, 1, 261, 60, 300, 23, 514, 1, 493, 105, 706, 1, 959, 1, 1066, 217, 1258, 1, 1927, 16, 2010, 301, 2545, 1, 3442, 64, 3898, 494, 4568, 1, 6555, 1, 6845, 841, 8610

Offset: 1

Omar E. Pol, Nov 22 2009

Row sums of triangle A168021 except the first column.

Row sums of triangle A168016 except the last column.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000041, A001065, A047968, A168016, A168017, A168018, A168020, A168021.

A047968 := proc(n) add(combinat[numbpart](d), d= numtheory[divisors](n) ) ; end proc: A000041 := proc(n) combinat[numbpart](n) ; end proc: A168111 := proc(n) A047968(n)-A000041(n) ; end proc: seq(A168111(n),n=1..90) ; # R. J. Mathar, Jan 25 2010

a[ n_] := If[n < 1, 0, Sum[ PartitionsP[ d] Boole[ d < n], {d, Divisors @ n}]]; (* Michael Somos, Feb 24 2014 *)

A168111(n) = sumdiv(n,d,(dAntti Karttunen, Nov 14 2017

a(n) = A047968(n) - A000041(n).
G.f.: Sum_{n > 0} A000041(n)*x^(2*n)/(1-x^n). - Mircea Merca, Feb 24 2014
G.f.: x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 8*x^8 + 4*x^9 + 10*x^10 + x^11 + ... - Michael Somos, Feb 24 2014

Terms beyond a(12) from R. J. Mathar, Jan 25 2010

New name from Omar E. Pol, Feb 25 2014

A168014 Sum of all parts of all partitions of n into equal parts that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 2, 3, 8, 5, 18, 7, 24, 18, 30, 11, 60, 13, 42, 45, 64, 17, 90, 19, 100, 63, 66, 23, 168, 50, 78, 81, 140, 29, 210, 31, 160, 99, 102, 105, 288, 37, 114, 117, 280, 41, 294, 43, 220, 225, 138, 47, 432, 98, 250, 153, 260, 53, 378

Offset: 0

Omar E. Pol, Nov 20 2009

Sum of all the parts in the partitions of n into two parts such that the smaller part divides the larger. - Wesley Ivan Hurt, Dec 22 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000005, A000041, A032741, A168015, A168020, A168021.

A168014:=n->n*(numtheory[tau](n)-1); seq(A168014(k), k=0..60); # Wesley Ivan Hurt, Oct 11 2013

Table[n*(DivisorSigma[0, n] - 1), {n, 0, 100}] (* Wesley Ivan Hurt, Oct 11 2013 *)

a(n) = n * A032741(n).

a(n) = n(d(n) - 1) = n*(A000005(n) - 1) = A038040(n) - n, if n >= 1. - Omar E. Pol, Jan 18 2013

a(14)-a(55) from Vincenzo Librandi, May 25 2010

New name from Omar E. Pol, Jan 18 2013

A168015 a(n) = A000041(n) + n*A032741(n).

Original entry on oeis.org

1, 1, 4, 6, 13, 12, 29, 22, 46, 48, 72, 67, 137, 114, 177, 221, 295, 314, 475, 509, 727, 855, 1068, 1278, 1743, 2008, 2514, 3091, 3858, 4594, 5814, 6873, 8509, 10242, 12412, 14988, 18265, 21674, 26129, 31302, 37618, 44624, 53468, 63304, 75395, 89359

Offset: 0

Omar E. Pol, Nov 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000005, A032741, A000041, A168014, A168020, A168021.

Table[PartitionsP[n] + n*(DivisorSigma[0, n] - 1), {n, 0, 50}] (* G. C. Greubel, Jul 05 2016 *)

a(n) = if (n, numbpart(n) + n*(numdiv(n)-1), numbpart(0)); \\ Michel Marcus, Jul 06 2016

a(14)-a(46) from Vincenzo Librandi, May 26 2010

cp's OEIS Frontend

A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

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A168016 Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.

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A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.

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A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

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A168019 Square array A(n,k) read by antidiagonals, in which row n lists the number of partitions of n into parts divisible by k+1.

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A168111 Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

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A168014 Sum of all parts of all partitions of n into equal parts that do not contain 1 as a part.

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