A339886 -id:A339886 - cp's OEIS frontend

A087897 Number of partitions of n into odd parts greater than 1.

1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756

Offset: 0

N. J. A. Sloane, Dec 04 2003

Also number of partitions of n into distinct parts which are not powers of 2.
Also number of partitions of n into distinct parts such that the two largest parts differ by 1.
Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006
Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007
In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013
Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014
From Gus Wiseman, Aug 22 2021: (Start)
Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:
  32  222  322  332   432   3322   3332   4332    4432    5432     43332
                2222  3222  22222  4322   33222   33322   33332    44322
                                   32222  222222  43222   43322    333222
                                                  322222  332222   432222
                                                          2222222  3222222
(End)

			1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...
q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...
a(10)=2 because we have [7,3] and [5,5].
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:
01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]
04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]
05:  [ 1 1 1 2 5 5 5 2 1 1 1 ]
06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]
07:  [ 1 1 3 5 5 5 3 1 1 ]
08:  [ 1 1 7 7 7 1 1 ]
09:  [ 1 2 2 5 5 5 2 2 1 ]
10:  [ 1 4 5 5 5 4 1 ]
11:  [ 2 2 2 2 3 3 3 2 2 2 2 ]
12:  [ 2 3 5 5 5 3 2 ]
13:  [ 2 7 7 7 2 ]
(End)
From _Gus Wiseman_, Feb 16 2021: (Start)
The a(7) = 1 through a(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.
  7  53  9    55  B    75    D    77    F      97    H      99      J
         333  73  533  93    553  95    555    B5    755    B7      775
                       3333  733  B3    753    D3    773    D5      955
                                  5333  933    5533  953    F3      973
                                        33333  7333  B33    5553    B53
                                                     53333  7533    D33
                                                            9333    55333
                                                            333333  73333
(End)

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from Alois P. Heinz)
C. Ballantine and M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi.
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.
R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

The ordered version is A000931.
Partitions with no ones are counted by A002865, ranked by A005408.
The even version is A035363, ranked by A066207.
The version for factorizations is A340101.
Partitions whose only even part is the smallest are counted by A341447.
The Heinz numbers of these partitions are given by A341449.
A000009 counts partitions into odd parts, ranked by A066208.
A025147 counts strict partitions with no 1's.
A025148 counts strict partitions with no 1's or 2's.
A026804 counts partitions whose smallest part is odd, ranked by A340932.
A027187 counts partitions with even length/maximum, ranks A028260/A244990.
A027193 counts partitions with odd length/maximum, ranks A026424/A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A058696 counts partitions of even numbers, ranked by A300061.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000041, A003114, A039900, A160786, A257991/A257992, A264396, A300272, A339662, A339737, A339886.

a087897 = p [3,5..] where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

To get 128 terms: t4 := mul((1+x^(2^n)),n=0..7); t5 := mul((1+x^k),k=1..128): t6 := series(t5/t4,x,100); t7 := seriestolist(t6);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013

max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&OddQ[Times@@#]&]],{n,0,30}] (* Gus Wiseman, Feb 16 2021 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A087897_T(n,k):
    if n==0: return 1
    if k<3 or n<0: return 0
    return A087897_T(n,k-2)+A087897_T(n-k,k)
def A087897(n): return A087897_T(n,n-(n&1^1)) # Chai Wah Wu, Sep 23 2023, after Alois P. Heinz

Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.
Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.
G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012
G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).
G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).
G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016
G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021
G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
a(2*n+1) = Sum{j>=1} A008284(n+1-j,2*j - 1) and a(2*n) = Sum{j>=1} A008284(n-j, 2*j). - Gregory L. Simay, Sep 22 2023

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset: 0

Gus Wiseman, Sep 30 2023

			The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row sums are A000041.
Diagonal k = n-1 is A002865.
Column k = 1 is A126796 (complete partitions), ranks A325781.
Central diagonal n = 2k is A126796 also.
For parts instead of sums we have A339737, rank stat A339662, min A257993.
This is the triangle for the rank statistic A365920.
Latter row sums are A365924 (incomplete partitions), ranks A365830.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
A366128 gives the least non-subset-sum of prime indices.
Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 3, 2, 4, 0, 5, 3, 4, 0, 6, 0, 7, 0, 5, 4, 8, 0, 5, 5, 5, 3, 9, 0, 10, 0, 6, 6, 6, 0, 11, 7, 7, 0, 12, 0, 13, 4, 6, 8, 14, 0, 7, 5, 8, 5, 15, 0, 7, 0, 9, 9, 16, 0, 17, 10, 7, 0, 8, 4, 18, 6, 10, 6, 19, 0, 20, 11, 7, 7, 8, 5, 21, 0, 7, 12

Offset: 1

Gus Wiseman, Sep 30 2023

nonn

This is the greatest element of {0,...,A056239(n)} that is not equal to A056239(d) for any divisor d|n, d>1. This definition is analogous to the Frobenius number of a numerical semigroup (see link), but it looks only at submultisets of a finite multiset, not all multisets of elements of a set.

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.

			The prime indices of 156 are {1,1,2,6}, with subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, so a(156) = 5.

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For binary indices instead of sums we have A063250.
Positions of first appearances > 2 are A065091.
Zeros are A325781, nonzeros A325798.
For prime indices instead of sums we have A339662, minimum A257993.
For least instead of greatest non-subset-sum we have A366128.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A079068, A098743, A264401, A286469 or A286470, A339737, A339886.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]

A366128 Least non-subset-sum of the multiset of prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 3, 1, 2, 1, 0, 1, 2, 1, 3, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 4, 1, 3, 1, 2, 1, 0, 1, 2, 1, 3, 1, 4, 1, 0, 1, 2, 1, 0, 1, 2, 1

Offset: 1

Gus Wiseman, Oct 06 2023

nonn

Least positive integer up to the sum of prime indices of n that is not the sum of prime indices of any divisor of n, or 0 if none exists.

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.

			The prime indices of 3906 are {1,2,2,4,11}, with least non-subset-sum 10, so a(3906) = 10.

Positions of ones are A005408.
Positions of twos appear to be A091999.
Zeros are A325781, nonzeros A325798.
For greatest instead of least we have A365920 (Frobenius number).
The triangle for this rank statistic is A365921 (partitions with least non-subset-sum k).
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A079068, A257993, A286469, A286470, A339662, A339886.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[If[nmz[prix[n]]=={},0,Min@@nmz[prix[n]]],{n,100}]

cp's OEIS Frontend

A087897 Number of partitions of n into odd parts greater than 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A339662 Greatest gap in the partition with Heinz number n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A366128 Least non-subset-sum of the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica