A066633 -id:A066633 - cp's OEIS frontend

A096541 Number of parts unequal to 1 in all partitions of the integer n. Also the difference between the labeled and the unlabeled case of one-element transitions from the partitions of n to the partitions of n+1.

0, 0, 1, 2, 5, 8, 16, 24, 41, 61, 95, 136, 204, 284, 407, 560, 779, 1050, 1432, 1901, 2543, 3338, 4393, 5698, 7411, 9513, 12226, 15562, 19803, 24993, 31538, 39506, 49456, 61546, 76499, 94603, 116858, 143679, 176431, 215802, 263576, 320796, 389900

Offset: 0

Thomas Wieder, Jun 24 2004

nonn

Also column 2 of A181187. - Omar E. Pol, Feb 18 2012

Sum over all partitions of n of the difference between the number of parts and the number of distinct parts. - Alois P. Heinz, Nov 18 2020

			The partitions of n=5 are [11111], [1112], [113], [122], [23], [14], [5] and they contain 0 + 1 + 1 + 2 + 2 + 1 + 1 = 8 = A096541(5) parts unequal to 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A066633, A093694, A093695, A094533, A006128, A339011.

main := proc(n::integer) local a,ndxp,ndxprt,ListOfPartitions,iverbose; with(combinat): ListOfPartitions:=partition(n); a:=0; for ndxp from 1 to nops(ListOfPartitions) do for ndxprt from 1 to nops(ListOfPartitions[ndxp]) do if op(ndxprt,ListOfPartitions[ndxp]) <> 1 then a := a + 1; fi; end do; end do; print("n, a(n):",n,a); end proc;
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 04 2012

f[n_] := Block[{l = Sort[ Flatten[ IntegerPartitions[n]]]}, Length[l] - Count[l, 1]]; Table[ f[n], {n, 0, 20}] (* Robert G. Wilson v, Jun 30 2004 *)
a[n_] := Sum[(DivisorSigma[0, k] - 1)*PartitionsP[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jan 14 2013, after Vladeta Jovovic *)

a(n)=sum(k=1,n,(numdiv(k)-1)*numbpart(n-k)) \\ Charles R Greathouse IV, Jan 14 2013

a(n) = A093694(n) - A000070(n).
a(n) = Sum_{k=1..n} (tau(k)-1)*numbpart(n-k). - Vladeta Jovovic, Jun 26 2004
a(n) ~ exp(Pi*sqrt(2*n/3))*(2*gamma - 2 + log(6*n/Pi^2))/(4*Pi*sqrt(2*n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 24 2016
a(n) = Sum_{i=1..floor(n/2)} A066633(n-i,i). - George Beck, Feb 15 2020
G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Robert G. Wilson v, Jun 30 2004

A221650 Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 3, 2, 2, 1, 0, 1, 1, 1, 0, 1, 5, 3, 3, 2, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 1, 7, 5, 5, 3, 0, 3, 2, 2, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 11, 7, 7, 5, 0, 5, 3, 3, 0, 3, 2, 0, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 21 2013

This tetrahedron shows a connection between divisors and partitions.
Conjecture 1: P(n,j,k) is the number of partitions of n that contain at least m parts of size k, where m = j/k, if k divides j otherwise P(n,j,k) = 0.
Conjecture 2: P(n,j,k) is the number of parts that are the m-th part of size k in all partitions of n, where m = j/k, if k divides j otherwise P(n,j,k) = 0.
The sum of all elements of slice n is A006128(n).
The sum of row j of slice n is A221530(n,j).
The sum of column k of slice n is A066633(n,k).
See also the tetrahedron of A221649.

			First six slices of tetrahedron are
---------------------------------------------------
n  j    k: 1  2  3  4  5  6      A221530   A006128
---------------------------------------------------
1  1       1,                       1         1
...................................................
2  1       1,                       1
2  2       1, 1,                    2         3
...................................................
3  1       2,                       2
3  2       1, 1,                    2
3  3       1, 0, 1,                 2         6
...................................................
4  1       3,                       3
4  2       2, 2,                    4
4  3       1, 0, 1,                 2
4  4       1, 1, 0, 1,              3        12
...................................................
5  1       5,                       5
5  2       3, 3,                    6
5  3       2, 0, 2,                 4
5  4       1, 1, 0, 1,              3
5  5       1, 0, 0, 0, 1,           2        20
...................................................
6  1       7,                       7
6  2       5, 5,                   10
6  3       3, 0, 3,                 6
6  4       2, 2, 0, 2,              6
6  5       1, 0, 0, 0, 1,           2
6  6       1, 1, 1, 0, 0, 1         4        35
...................................................

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows n = 1..40 of the tetrahedron, flattened)

Cf. A000005, A006128, A027750, A051731, A066633, A127093, A221530, A221649.

A221650row[n_]:=Flatten[Table[If[Divisible[j,k],PartitionsP[n-j],0],{j,n},{k,j}]];Array[A221650row,10] (* Paolo Xausa, Sep 26 2023 *)

P(n,j,k) = A051731(j,k)*A000041(n-j) = (1/k)*A221649(n,j,k).

A221876 T(n,k) is the number of order-preserving full contraction mappings (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 12, 5, 2, 1, 28, 12, 5, 2, 1, 64, 28, 12, 5, 2, 1, 144, 64, 28, 12, 5, 2, 1, 320, 144, 64, 28, 12, 5, 2, 1, 704, 320, 144, 64, 28, 12, 5, 2, 1, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1

Offset: 1

Abdullahi Umar, Feb 28 2013

Row sum is A001792(n-1).
The matrix inverse starts
1;
-2,1;
-1,-2,1;
0,-1,-2,1;
1,0,-1,-2,1;
2,1,0,-1,-2,1;
3,2,1,0,-1,-2,1;
4,3,2,1,0,-1,-2,1;
5,4,3,2,1,0,-1,-2,1;
6,5,4,3,2,1,0,-1,-2,1;
7,6,5,4,3,2,1,0,-1,-2,1; - R. J. Mathar, Apr 12 2013
...
T(n,k) is also the total number of occurrences of parts k in all compositions (ordered partitions) of n, see example. The equivalent sequence for partitions is A066633. Omar E. Pol, Aug 26 2013

			T(5,3) = 5 because there are exactly 5 order-preserving full contraction mappings (of a 5-chain) with exactly 3 fixed points, namely: (12333), (12334), (22344), (23345), (33345).
Triangle begins:
1,
2, 1,
5, 2, 1,
12, 5, 2, 1,
28, 12, 5, 2, 1,
64, 28, 12, 5, 2, 1,
144, 64, 28, 12, 5, 2, 1,
320, 144, 64, 28, 12, 5, 2, 1,
704, 320, 144, 64, 28, 12, 5, 2, 1,
1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
...
Note that column k is column 1 shifted down by k positions.
Row 4 is [12, 5, 2, 1]: in the compositions of 4
[ 1]  [ 1 1 1 1 ]
[ 2]  [ 1 1 2 ]
[ 3]  [ 1 2 1 ]
[ 4]  [ 1 3 ]
[ 5]  [ 2 1 1 ]
[ 6]  [ 2 2 ]
[ 7]  [ 3 1 ]
[ 8]  [ 4 ]
there are 12 parts=1, 5 parts=2, 2 part=3, and 1 part=4.
- _Joerg Arndt_, Sep 01 2013

A. D. Adeshola, V. Maltcev and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, arXiv:1303.7428 [math.CO], 2013.
A. D. Adeshola, A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, JMCC 106 (2017) 37-49

Cf. A001792, A221877, A221878, A221879, A221880, A221881, A221882.

T[n_, n_] = 1; T[n_, k_] := (n - k + 3)*2^(n - k - 2);
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 21 2018 *)

T(n,n) = 1, T(n,k) = (n-k+3)*2^(n-k-2) for n>=2 and n > k > 0.
T(2*n+1,n+1) = T(n+1,1) = A045623(n) for n>=0.
T(n,k) = A045623(n-k), n>=1, 1<=k<=n. - Omar E. Pol, Sep 01 2013

A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

Original entry on oeis.org

2, 4, 2, 10, 4, 2, 20, 8, 4, 2, 38, 16, 8, 4, 2, 68, 30, 16, 8, 4, 2, 118, 52, 28, 16, 8, 4, 2, 196, 88, 48, 28, 16, 8, 4, 2, 318, 144, 82, 48, 28, 16, 8, 4, 2, 504, 230, 132, 80, 48, 28, 16, 8, 4, 2, 782, 360, 208, 128, 80, 48, 28, 16, 8, 4, 2, 1192, 552, 324, 202, 128, 80, 48, 28, 16, 8, 4, 2

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that row n lists the first differences of row n of triangle A235797 together with 2 (as the final term of the row).

The equivalent sequence for partitions is A066633.

			Triangle begins:
2;
4,   2;
10,  4,  2;
20,  8,  4,  2;
38, 16,  8,  4,  2;
68, 30, 16,  8,  4,  2;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums give A235792.

Cf. A015128, A066633, A235790, A235793, A235797, A236000, A236001.

A(n)={my(p=prod(k=1, n, (1 + x^k)/(1 - x^k) + O(x*x^n))); Mat(vector(n, k, Col(2*(p + O(x*x^(n-k)))*x^k/((1 - x^k)*(1 + x^k)), -n)))}
{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

G.f. of column k: 2*(x^k/((1 - x^k)*(1 + x^k))) * Product_{j>0} (1 + x^j)/(1 - x^j). - Andrew Howroyd, Feb 19 2020

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 8, 8, 32, 144, 432, 2160, 27000, 582120, 7623000, 336936600, 6740402760, 543454231320, 57619849046760, 4683793138766280, 412882704970215480, 88171665744392750520, 12780536107937124847320, 2685589660883755945879560, 942036670625665177379096280

Offset: 0

Gus Wiseman, May 07 2019

nonn

Also the Heinz number of row n of A015716 (with zeros removed).

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.
The sequence of terms together with their prime indices begins:
               1: {}
               2: {1}
               2: {1}
               8: {1,1,1}
               8: {1,1,1}
              32: {1,1,1,1,1}
             144: {1,1,1,1,2,2}
             432: {1,1,1,1,2,2,2}
            2160: {1,1,1,1,2,2,2,3}
           27000: {1,1,1,2,2,2,3,3,3}
          582120: {1,1,1,2,2,2,3,4,4,5}
         7623000: {1,1,1,2,2,3,3,3,4,5,5}
       336936600: {1,1,1,2,2,3,3,4,5,5,6,7}
      6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}
    543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}
  57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

Alois P. Heinz, Table of n, a(n) for n = 0..172

Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

b:= proc(n, i) option remember;
      `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+
          (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(
    coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 23 2024

Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

a(n) = A181819(A003963(A325505(n))).

A056239(a(n)) = A015723(n).

A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 20 2019

Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with 1.

Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, 1, 1, (n - 3)/2 zeros, 1.

			Triangle begins:
1;
0, 1;
1, 1, 1;
0, 0, 0, 1;
0, 1, 1, 0, 1;
1, 1, 1, 0, 0, 1;
0, 0, 1, 1, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 1;
0, 1, 1, 2, 1, 0, 0, 0, 1;
1, 1, 1, 1, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1;
0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1;
0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 1, 1, 2, 1, 0, 0, 0, 1].

Row sums give A204217.
Column 1 gives A010054, n >= 1.
Leading diagonal gives A000012.
Cf. A000079, A001227, A065091, A066633, A237048, A237593, A245579, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328362.

A052810 a(n) = 1 + (number of partitions of n, n>0).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 23, 31, 43, 57, 78, 102, 136, 177, 232, 298, 386, 491, 628, 793, 1003, 1256, 1576, 1959, 2437, 3011, 3719, 4566, 5605, 6843, 8350, 10144, 12311, 14884, 17978, 21638, 26016, 31186, 37339, 44584, 53175, 63262, 75176, 89135

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n>0: number of occurrences of n in partitions of 2*n: a(n)=A066633(2*n,n), cf. A058696. - Reinhard Zumkeller, Feb 22 2004

Paolo Xausa, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 772.

Cf. A000041, A057427.

spec := [S,{B=Set(C),C=Sequence(Z,1 <= card), S = Union(C,B)}, unlabeled]:
seq(combstruct[count](spec, size=n), n=0..20);
A052810 := n -> combinat:-numbpart(n) + ifelse(n=0, 0, 1):
seq(A052810(i), i=0..50);

Join[{1}, PartitionsP[Range[50]] + 1] (* Paolo Xausa, Jun 21 2024 *)

G.f.: exp(Sum_{j >= 1} (x^j)/(1 - x^j)/j) - x/(x - 1). [Simplified by Paolo Xausa, Jun 21 2024]

a(n) = A000041(n) + A057427(n). - Alois P. Heinz, May 14 2023

Better description and more terms from Vladeta Jovovic, Oct 06 2001

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 1, 1, 1, 2, 0, 4, 1, 0, 3, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| 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 |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A372890 Sum of binary ranks of all integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

0, 1, 4, 10, 25, 52, 115, 228, 471, 931, 1871, 3687, 7373, 14572, 29049, 57694, 115058, 229101, 457392, 912469, 1822945, 3640998, 7277426, 14544436, 29079423, 58137188, 116254386, 232465342, 464889800, 929691662, 1859302291, 3718428513, 7436694889, 14873042016

Offset: 0

Gus Wiseman, May 23 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4 with sum 25, so a(4) = 25.

Alois P. Heinz, Table of n, a(n) for n = 0..3321

For Heinz number (not binary rank) we have A145519, row sums of A215366.
For Heinz number the strict version is A147655, row sums of A246867.
The strict version is A372888, row sums of A118462.
A005117 gives Heinz numbers of strict integer partitions.
A048675 gives binary rank of prime indices, distinct A087207.
A061395 gives greatest prime index, least A055396.
A118457 lists strict partitions in Mathematica order.
A277905 groups all positive integers by binary rank of prime indices.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A005940, A018819, A019565, A066633, A225620, A344086, A365410.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p->[0, p[1]*2^(i-1)]+p)(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 23 2024

Table[Total[Total[2^(#-1)]&/@IntegerPartitions[n]],{n,0,10}]

From Alois P. Heinz, May 23 2024: (Start)
a(n) = Sum_{k=1..n} 2^(k-1) * A066633(n,k).
a(n) mod 2 = A365410(n-1) for n>=1. (End)

A325500 Heinz number of the set of Heinz numbers of integer partitions of n. Heinz numbers of rows of A215366.

Original entry on oeis.org

2, 3, 35, 2717, 22235779, 3163570326979, 51747966790650260753033, 188828800892079861898153036258130093, 2034903808706825942766196978067005215014684343665351270467, 75367279796373180679613801327275978589820813788234346991420766634058571423774287454563

Offset: 0

Gus Wiseman, May 05 2019

nonn

The Heinz number of a set of positive integers {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

All terms are squarefree and pairwise relatively prime.

			The integer partitions of 3 are {(3), (2,1), (1,1,1)}, with Heinz numbers {5,6,8}, with Heinz number prime(5)*prime(6)*prime(8) = 2717, so a(3) = 2717.
The sequence of terms together with their prime indices begins:
                        2: {1}
                        3: {2}
                       35: {3,4}
                     2717: {5,6,8}
                 22235779: {7,9,10,12,16}
            3163570326979: {11,14,15,18,20,24,32}
  51747966790650260753033: {13,21,22,25,27,28,30,36,40,48,64}

Index entries for sequences related to Heinz numbers

A subsequence of A005117.

Cf. A001222, A002110, A003963, A006128, A007870, A056239, A066186, A066633, A112798, A145519, A215366, A325501, A325505, A325507.

Table[Times@@Prime/@(Times@@Prime/@#&/@IntegerPartitions[n]),{n,0,5}]

A001221(a(n)) = A001222(a(n)) = A000041(n).
A056239(a(n)) = A145519(n).
A003963(a(n)) = A325501(n).
A181819(A003963(a(n))) = A325507(n).

cp's OEIS Frontend

A096541 Number of parts unequal to 1 in all partitions of the integer n. Also the difference between the labeled and the unlabeled case of one-element transitions from the partitions of n to the partitions of n+1.

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A221650 Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.

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A221876 T(n,k) is the number of order-preserving full contraction mappings (of an n-chain) with exactly k fixed points.

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A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

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A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

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A328361 Triangle read by rows: T(n,k) is the total number of k's in all partitions of n into consecutive parts, (1 <= k <= n).

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A052810 a(n) = 1 + (number of partitions of n, n>0).

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