A053445 -id:A053445 - cp's OEIS frontend

A083751 Number of partitions of n into >= 2 parts and with minimum part >= 2.

0, 0, 0, 1, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset: 1

Jon Perry, Jun 17 2003

nonn

Also number of partitions of n such that the largest part is at least 2 and occurs at least twice. Example: a(6)=3 because we have [3,3],[2,2,2] and [2,2,1,1]. - Emeric Deutsch, Mar 29 2006
Also number of partitions of n that contain emergent parts (Cf. A182699). - Omar E. Pol, Oct 21 2011
Also number of regions in the last section of the set of partitions of n that do not contain 1 as a part (cf. A187219). - Omar E. Pol, Mar 04 2012
Schneider calls these "nuclear partitions" and gives a remarkable formula relating a(n), the number of partitions of n, and a sum over the two greatest parts of each such partition. - Charles R Greathouse IV, Dec 04 2019

			a(6) = 3, as 6 = 2+4 = 3+3 = 2+2+2.
a(6) = 3 because 6 = 2+4 = 3+3 = 2+2+2.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Robert Schneider, Nuclear partitions and a formula for p(n), arXiv:1912.00575 [math.NT], 2019.

Cf. A053445, A072380, A008483, A026796, A035989, A036000, A002865, A081094.

First differences of A000094.

g:=sum(x^(2*j)/product(1-x^i,i=1..j),j=2..50): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Mar 29 2006

Drop[CoefficientList[Series[1/Product[(1-x^k)^1, {k, 2, 50}], {x, 0, 50}], x]-1, 2]
(* or *) Table[Count[IntegerPartitions[n], q_List /; Length[q] > 1 && Min[q] >= 2 ], {n, 24}]

a(n) = A000041(n) - A000041(n-1) - 1, n > 1. - Vladeta Jovovic, Jun 18 2003
G.f.: Sum_{j>=2} x^(2j)/Product_{i=1..j} (1-x^i). - Emeric Deutsch, Mar 29 2006
a(n) = A002865(n) - 1, n > 1. - Omar E. Pol, Oct 21 2011
a(n) = A187219(n) - 1. - Omar E. Pol, Mar 04 2012

More terms from Vladeta Jovovic and Wouter Meeussen, Jun 18 2003

Description corrected by James Sellers, Jun 21 2003

A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, -6, 45, -50, 113, -98, 73, 274, -1159, 3563, -8707, 19024, -36977, 64582, -98401, 121436, -81961, -147383, 860871, -2709964, 7110655, -17077217, 38873213, -85085216, 179965720, -367884935, 725051361, -1372311916, 2481473639, -4257624155

Offset: 0

Gus Wiseman, Oct 22 2024

sign

These are the row-sums of the triangle-version of A377051.

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = -6.

The version for primes is A140119, noncomposites A376683, composites A377034.
For squarefree numbers we have A377039, nonsquarefree A377047.
These are the antidiagonal-sums of A377051.
The unsigned version is A377053.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A093555, A174965, A246655, A361102, A376340, A376596, A376598.

nn=20;
t=Table[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset: 0

Gus Wiseman, Dec 13 2024

			As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
For primes we have A095195 or A376682.
For partitions we have A175804.
First column is A293467 (up to sign).
For composites we have A377033.
For squarefree numbers we have A377038.
For nonsquarefree numbers we have A377046.
For prime powers we have A377051.
Position of first zero in each row is A377285.
Triangle's row-sums are A378970, absolute A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A225485 or A325280.

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, 24, 45, 80, 123, 174, 229, 382, 1219, 3591, 8849, 19288, 37899, 67442, 108323, 156054, 206733, 311525, 860955, 2710374, 7111657, 17080759, 38884849, 85124764, 180097856, 368321633, 726482493, 1377039690, 2496856437, 4306569569, 7016267449

Offset: 0

Gus Wiseman, Oct 22 2024

nonn

These are the row-sums of the absolute value of the triangle-version of A377051.

			The sixth antidiagonal of A377051 is (8, 1, -1, -2, -3, -4, -5), so a(6) = 24.

The version for primes is A376681, noncomposites A376684, composites A377035.
For squarefree numbers we have A377040, nonsquarefree A377048.
This is the antidiagonal-sums of the absolute value of A377051.
The signed version is A377052.
For leaders we have A377054, for primes A007442 or A030016.
For first zero-positions we have A377055.
A version for partitions is A377056, cf. A175804, A053445, A281425, A320590.
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
Cf. A000961, A025475, A053707, A057820, A069623, A093555, A174965, A246655, A361102, A376340, A376596.

nn=20;
t=Table[Differences[NestList[NestWhile[#+1&, #+1,!PrimePowerQ[#]&]&,1,2*nn],k],{k,0,nn}];
Total/@Abs[Table[t[[j,i-j+1]],{i,nn},{j,i}]]

A378972 Second differences of the strict partition numbers A000009.

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, 3, 2, 3, 4, 3, 4, 6, 4, 6, 8, 6, 9, 10, 9, 12, 14, 13, 16, 19, 18, 22, 26, 24, 30, 34, 34, 40, 45, 46, 53, 60, 62, 70, 79, 82, 93, 104, 108, 122, 136, 142, 160, 176, 186, 208, 228, 243, 268

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			The strict partition numbers begin (A000009):
  1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, ...
with differences (A087897 without first term):
  0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, ...
with differences (a(n)):
  0, 1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 1, 2, ...

For primes we have A036263.
The version for partitions is A053445.
For composites we have A073445.
For squarefree numbers we have A376590.
For nonsquarefree numbers we have A376593.
For powers of primes (inclusive) we have A376596.
For non powers of primes (inclusive) we have A376599.
Second row of A378622. See also:
- A293467 gives first column (up to sign).
- A377285 gives position of first zero in each row.
- A378970 gives row-sums.
- A378971 gives absolute value row-sums.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A225485, A325242, A325268.

Differences[Table[PartitionsQ[n],{n,0,100}],2]

A072380 Third differences of partition numbers A000041.

Original entry on oeis.org

-1, 1, -1, 2, -2, 3, -2, 3, -2, 5, -4, 7, -3, 7, -3, 11, -5, 15, -4, 17, -2, 24, -4, 32, 1, 38, 5, 53, 7, 70, 18, 86, 33, 115, 45, 152, 74, 191, 109, 254, 150, 331, 218, 420, 307, 551, 410, 716, 567, 913, 767, 1186, 1015, 1529, 1358, 1951, 1799, 2513, 2344, 3222, 3079, 4096, 4009, 5237, 5173

Offset: 0

N. J. A. Sloane, Apr 25 2003

sign

Comtet appears to say this is nonnegative, which is only true for n sufficiently large.

An explanation is given by Odlyzko. - Moshe Shmuel Newman, Jun 11 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.

Cf. A000041, A002865, A053445, A081094, A081095.

Differences[PartitionsP[Range[0,70]],3] (* Harvey P. Dale, Aug 16 2012 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (72 * sqrt(2) * n^(5/2)). - Vaclav Kotesovec, Oct 06 2017

A081094 4th differences of partition numbers A000041.

Original entry on oeis.org

2, -2, 3, -4, 5, -5, 5, -5, 7, -9, 11, -10, 10, -10, 14, -16, 20, -19, 21, -19, 26, -28, 36, -31, 37, -33, 48, -46, 63, -52, 68, -53, 82, -70, 107, -78, 117, -82, 145, -104, 181, -113, 202, -113, 244, -141, 306, -149, 346, -146, 419, -171, 514, -171, 593, -152, 714, -169, 878, -143, 1017, -87, 1228, -64, 1497

Offset: 0

N. J. A. Sloane, Apr 25 2003

sign

Comtet appears to say this is nonnegative, which is only true for n sufficiently large.

An explanation is given by Odlyzko. - Moshe Shmuel Newman, Jun 11 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.

Cf. A000041, A002865, A053445, A072380, A081095.

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^4 / (144 * sqrt(3) * n^3). - Vaclav Kotesovec, Oct 06 2017

A081095 5th differences of partition numbers A000041.

Original entry on oeis.org

-4, 5, -7, 9, -10, 10, -10, 12, -16, 20, -21, 20, -20, 24, -30, 36, -39, 40, -40, 45, -54, 64, -67, 68, -70, 81, -94, 109, -115, 120, -121, 135, -152, 177, -185, 195, -199, 227, -249, 285, -294, 315, -315, 357, -385, 447, -455, 495, -492, 565, -590, 685, -685, 764, -745, 866, -883, 1047, -1021, 1160

Offset: 0

N. J. A. Sloane, Apr 25 2003

sign

Comtet appears to say this is nonnegative, which is only true for n sufficiently large.

An explanation is given by Odlyzko. - Moshe Shmuel Newman, Jun 11 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.

Cf. A000041, A002865, A053445, A072380, A081094.

Differences[PartitionsP[Range[0,70]],5] (* Harvey P. Dale, Jul 27 2014 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^5 / (432 * sqrt(2) * n^(7/2)). - Vaclav Kotesovec, Oct 06 2017

A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 3, 11, 2, 36, -27, 142, -207, 595, -1066, 2497, -4878, 10726, -22189, 48383, -103318, 224296, -480761, 1030299, -2186942, 4626313, -9740648, 20492711, -43109372, 90843475, -191769296, 405528200, -858373221, 1817311451, -3845483855, 8129033837

Offset: 0

Gus Wiseman, Dec 12 2024

sign

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 3.

For primes we have A140119 or A376683, unsigned A376681 or A376684.
These are the antidiagonal-sums of A175804.
First column of the same array is A281425.
For composites we have A377034, unsigned A377035.
For squarefree numbers we have A377039, unsigned A377040.
For nonsquarefree numbers we have A377049, unsigned A377048.
For prime powers we have A377052, unsigned A377053.
The unsigned version is A378621.
The version for strict partitions is A378970 (row-sums of A378622), unsigned A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A116608, A225486, A293467, A325242, A325245, A325268, A377285.

nn=20;
t=Table[Differences[PartitionsP/@Range[0,2nn],k],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

cp's OEIS Frontend

A083751 Number of partitions of n into >= 2 parts and with minimum part >= 2.

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A377052 Antidiagonal-sums of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

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A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

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A377053 Antidiagonal-sums of the absolute value of the array A377051(n,k) = n-th term of k-th differences of powers of primes.

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A378972 Second differences of the strict partition numbers A000009.

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A072380 Third differences of partition numbers A000041.

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A081094 4th differences of partition numbers A000041.

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A081095 5th differences of partition numbers A000041.

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A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).