A365676 -id:A365676 - cp's OEIS frontend

A002133 Number of partitions of n with exactly two part sizes.

0, 0, 1, 2, 5, 6, 11, 13, 17, 22, 27, 29, 37, 44, 44, 55, 59, 68, 71, 81, 82, 102, 97, 112, 109, 136, 126, 149, 141, 168, 157, 188, 176, 212, 182, 231, 207, 254, 230, 266, 241, 300, 259, 319, 283, 344, 295, 373, 311, 386, 352, 417, 353, 452, 368, 460, 418, 492, 413

Offset: 1

N. J. A. Sloane

Also number of solutions to the Diophantine equation ab + bc + cd = n, with a,b,c >= 1. - N. J. A. Sloane, Jun 17 2011

A generalized sum of divisors function.

			a(8) = 13 because we have 71, 62, 611, 53, 5111, 422, 41111, 332, 3311, 311111, 22211, 221111, 2111111.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_3(n).
E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
W. J. Keith, Partitions into a small number of part sizes, Int. Jour. of Num. Thy., Vol 13 no. 1, 229-241 (2017), doi:10.1142/S1793042117500130
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
N. Benyahia Tani, S. Bouroubi, and O. Kihel, An effective approach for integer partitions using exactly two distinct sizes of parts, Bulletin du Laboratoire 03 (2015), 18-27.
N. B. Tani and S. Bouroubi, Enumeration of the Partitions of an Integer into Parts of a Specified Number of Different Sizes and Especially Two Sizes, J. Integer Seqs., Vol. 14 (2011), #11.3.6. (This sequence appears as the rightmost column of Table 1 on p. 10.)

A diagonal of A060177.

Cf. A002134.

g:=sum(sum(x^(i+j)/(1-x^i)/(1-x^j),j=1..i-1),i=1..80): gser:=series(g,x=0,65): seq(coeff(gser,x^n),n=1..60); # Emeric Deutsch, Mar 30 2006
with(numtheory); D00:=n->add(tau(j)*tau(n-j),j=1..n-1); L3:=n->(D00(n)+tau(n)-sigma(n))/2; [seq(L3(n),n=1..60)]; # N. J. A. Sloane, Jun 17 2011
A002133 := proc(n)
    A055507(n-1)+numtheory[tau](n)-numtheory[sigma](n) ;
    %/2 ;
end proc: # R. J. Mathar, Jun 15 2022
# Using function P from A365676:
A002133 := n -> P(n, 2, n): seq(A002133(n), n = 1..59); # Peter Luschny, Sep 15 2023

nn=50;ss=Sum[Sum[x^(i+j)/(1-x^i)/(1-x^j),{j,1,i-1}],{i,1,nn}];Drop[CoefficientList[Series[ss,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 13 2012 *)
Table[DivisorSigma[0, n] - DivisorSigma[1, n] + Sum[DivisorSigma[0, k]*DivisorSigma[0, n - k], {k, 1, n - 1}], {n, 1, 100}]/2 (* Vaclav Kotesovec, Aug 30 2025 *)

from sympy import divisor_count, divisor_sigma
def A002133(n): return sum(divisor_count(j)*divisor_count(n-j) for j in range(1,(n-1>>1)+1)) + ((divisor_count(n+1>>1)**2 if n-1&1 else 0)+divisor_count(n)-divisor_sigma(n)>>1) # Chai Wah Wu, Sep 15 2023

G.f.: Sum_{i>=1} Sum_{j=1..i-1} x^(i+j)/((1-x^i)*(1-x^j)). - Emeric Deutsch, Mar 30 2006
Andrews gives a formula which is programmed up in the Maple code below. - N. J. A. Sloane, Jun 17 2011
G.f.: (G(x)^2-H(x))/2 where G(x) = Sum_{k>0} x^k/(1-x^k) and H(x) = Sum_{k>0} x^(2*k)/(1-x^k)^2. More generally, we obtain g.f. for number of partitions of n with m types of parts if we substitute x(i) with -Sum_{k>0}(x^n/(x^n-1))^i in cycle index Z(S(m); x(1),x(2),...,x(m)) of symmetric group S(m) of degree m. - Vladeta Jovovic, Sep 18 2007

A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

Original entry on oeis.org

1, 2, 5, 10, 15, 25, 37, 52, 67, 97, 117, 154, 184, 235, 277, 338, 385, 469, 531, 630, 698, 810, 910, 1038, 1144, 1295, 1425, 1577, 1741, 1938, 2089, 2301, 2505, 2700, 2970, 3189, 3444, 3703, 4004, 4242, 4617, 4882, 5244, 5558, 5999, 6221, 6755, 7050, 7576

Offset: 6

N. J. A. Sloane

			a(8) = 5 because we have 5+2+1, 4+3+1, 4+2+1+1, 3+2+2+1, 3+2+1+1+1.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 6..10000
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

A diagonal of A060177.

Column k=3 of A116608. - Alois P. Heinz, Nov 07 2012

# Using function P from A365676:
A002134 := n -> P(n, 3, n): seq(A002134(n), n = 6..54); # Peter Luschny, Sep 15 2023

nn=40;sss=Sum[Sum[Sum[x^(i+j+k)/(1-x^i)/(1-x^j)/(1-x^k),{k,1,j-1}], {j,1,i-1}], {i,1,nn}]; Drop[CoefficientList[Series[sss,{x,0,nn}],x],6]  (* Geoffrey Critzer, Sep 13 2012 *)

G.f.: Sum_{i>=1} Sum_{j=1..i-1} Sum_{k=1..j-1} x^(i+j+k)/((1-x^i)*(1-x^j)* (1-x^k)). - Geoffrey Critzer, Sep 13 2012

Better description and more terms from Naohiro Nomoto, Jan 24 2002

More terms from Vladeta Jovovic, Nov 02 2003

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 2, 2, 4, 3, 3, 4, 4, 3, 6, 5, 8, 8, 13, 13, 16, 17, 21, 22, 25, 26, 32, 34, 37, 44, 47, 55, 62, 72, 78, 94, 103, 118, 132, 151, 163, 189, 205, 230, 251, 284, 307, 346, 377, 420, 462, 515, 562, 629, 690, 763

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(n) partitions for n = 1, 2, 10, 13, 14, 19, 20, 21:
  1  .  32221   332221   333221   4333321     43333211    43333221
        322111  333211   3322211  43322221    44322221    433332111
                3322111  3332111  433321111   433222211   443222211
                4321111           443221111   443321111   444321111
                                  543211111   4332221111  4332222111
                                  4322221111              4333221111
                                                          4432221111
                                                          5432211111

John Tyler Rascoe, Table of n, a(n) for n = 0..500

Without the middle statistic we have A000009, ranked by A055932.
Counting partitions by the LHS gives A008284 (strict A008289), rank statistic A061395.
Counting partitions by the middle statistic gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116608/A365676, rank statistic A001221.
Without the LHS we have A239964, ranked by A212166.
Without the RHS we have A240312, ranked by A381542.
The Heinz numbers of these partitions are listed by A381543.
A000041 counts integer partitions.
A047993 counts partitions with max part = length, ranks A106529.
A116598 counts ones in partitions, rank statistic A007814.
A381438 counts partitions by last part part of section-sum partition.
Cf. A047966, A130091, A237984, A239455, A241131, A351293, A362608, A363719, A381079, A381544, A382303.

Table[Length[Select[IntegerPartitions[n],Max@@#==Max@@Length/@Split[#]==Length[Union[#]]&]],{n,0,30}]

A_x(N) = {if(N<1,[0],my(x='x+O('x^(N+1))); concat([0],Vec(sum(i=1,N, prod(j=1,i, (x^j-x^((i+1)*j))/(1-x^j)) - prod(j=1,i, (x^j-x^(i*j))/(1-x^j))))))}
A_x(60) \\ John Tyler Rascoe, Mar 25 2025

G.f.: Sum_{i>0} (B(i+1,i,x) - B(i,i,x)) where B(a,c,x) = Product_{j=1..c} (x^j - x^(a*j))/(1 - x^j). - John Tyler Rascoe, Mar 25 2025

A365630 Number of partitions of n with exactly four part sizes.

Original entry on oeis.org

1, 2, 5, 10, 20, 30, 52, 77, 117, 162, 227, 309, 414, 535, 692, 873, 1100, 1369, 1661, 2030, 2438, 2925, 3450, 4108, 4759, 5570, 6440, 7457, 8491, 9798, 11020, 12593, 14125, 15995, 17820, 20074, 22182, 24833, 27379, 30422, 33351, 36996, 40346, 44445, 48336, 53048, 57494

Offset: 10

Seiichi Manyama, Sep 13 2023

nonn

			a(11) = 2 because we have 5+3+2+1, 4+3+2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 10..10000

A diagonal of A060177.
Column k=4 of A116608.
Cf. A000005, A002133, A002134, A365631.

# Using function P from A365676:
A365630 := n -> P(n, 4, n): seq(A365630(n), n = 10..56); # Peter Luschny, Sep 15 2023

from sympy.utilities.iterables import partitions
def A365630(n): return sum(1 for p in partitions(n) if len(p)==4) # Chai Wah Wu, Sep 14 2023

G.f.: Sum_{0

A365631 Number of partitions of n with exactly five part sizes.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 58, 95, 147, 222, 323, 462, 636, 889, 1184, 1584, 2060, 2686, 3403, 4353, 5433, 6768, 8319, 10230, 12363, 15011, 17943, 21467, 25403, 30044, 35231, 41294, 48002, 55718, 64328, 74086, 84880, 97071, 110607, 125692, 142313, 160728, 181112, 203438, 228124

Offset: 15

Seiichi Manyama, Sep 13 2023

nonn

			a(16) = 2 because we have 6+4+3+2+1, 5+4+3+2+1+1.

Alois P. Heinz, Table of n, a(n) for n = 15..5000

A diagonal of A060177.
Column k=5 of A116608.
Cf. A000005, A002133, A002134, A365630.
Cf. A364809.

# Using function P from A365676:
A365631 := n -> P(n, 5, n): seq(A365631(n), n = 15..59); # Peter Luschny, Sep 15 2023

from sympy.utilities.iterables import partitions
def A365631(n): return sum(1 for p in partitions(n) if len(p)==5) # Chai Wah Wu, Sep 14 2023

G.f.: Sum_{0

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Original entry on oeis.org

2, 12, 18, 36, 120, 270, 360, 540, 600, 750, 1080, 1350, 1500, 1680, 1800, 2250, 2700, 3000, 4500, 5040, 5400, 5670, 6750, 8400, 9000, 11340, 11760, 13500, 15120, 22680, 25200, 26250, 27000, 28350, 35280, 36960, 39690, 42000, 45360, 52500, 56700, 58800, 72030

Offset: 1

Gus Wiseman, Mar 24 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms together with their prime indices begin:
      2: {1}
     12: {1,1,2}
     18: {1,2,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    270: {1,2,2,2,3}
    360: {1,1,1,2,2,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    750: {1,2,3,3,3}
   1080: {1,1,1,2,2,2,3}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   1680: {1,1,1,1,2,3,4}
   1800: {1,1,1,2,2,3,3}

Counting partitions by the LHS gives A008284, rank statistic A061395.
Without the RHS we have A055932, counted by A000009.
Counting partitions by the RHS gives A091602, rank statistic A051903.
Counting partitions by the middle statistic gives A116608/A365676, rank stat A001221.
Without the LHS we have A212166, counted by A239964.
Without the middle statistic we have A381542, counted by A240312.
Partitions of this type are counted by A382302.
A000040 lists the primes, differences A001223.
A001222 counts prime factors, distinct A001221.
A047993 counts balanced partitions, ranks A106529.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents partition conjugation in terms of Heinz numbers.
Cf. A000720, A048767, A062457, A066328, A130091, A317090, A363719, A363740, A380955, A381632.

Select[Range[2,1000],PrimePi[FactorInteger[#][[-1,1]]]==PrimeNu[#]==Max@@FactorInteger[#][[All,2]]&]

A061395(a(n)) = A001221(a(n)) = A051903(a(n)).

Showing 1-7 of 7 results.

cp's OEIS Frontend

A365675 a(n) = Sum_{k=0..n} p(k) where the p(k) are the partial sums of row n of A365676.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A002133 Number of partitions of n with exactly two part sizes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A002134 Generalized divisor function. Number of partitions of n with exactly three part sizes.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A382302 Number of integer partitions of n with greatest part, greatest multiplicity, and number of distinct parts all equal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A365630 Number of partitions of n with exactly four part sizes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

A365631 Number of partitions of n with exactly five part sizes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Python

Formula

A381543 Numbers > 1 whose greatest prime index (A061395), number of distinct prime factors (A001221), and greatest prime multiplicity (A051903) are all equal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula