A063834 -id:A063834 - cp's OEIS frontend

A119441 Distribution of A063834 in Abramowitz and Stegun order.

1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21

Offset: 1

Alford Arnold, May 19 2006

			1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A063834, A119442, A000041 (row lengths and also first column)

# Compare two partitions (list) in AS order.
AScompare := proc(p1,p2)
    if nops(p1) > nops(p2) then
        return 1;
    elif nops(p1) < nops(p2) then
        return -1;
    else
        for i from 1 to nops(p1) do
            if op(i,p1) > op(i,p2) then
                return 1;
            elif op(i,p1) < op(i,p2) then
                return -1;
            end if;
        end do:
        return 0 ;
    end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
    local pi,insrt,p,ex ;
    pi := [] ;
    for p in combinat[partition](n) do
        insrt := 0 ;
        for ex from 1 to nops(pi) do
            if AScompare(p, op(ex,pi)) > 0 then
                insrt := ex ;
            end if;
        end do:
        if nops(pi) = 0 then
            pi := [p] ;
        elif insrt = 0 then
            pi := [p,op(pi)] ;
        elif insrt = nops(pi) then
            pi := [op(pi),p] ;
        else
            pi := [op(1..insrt,pi),p,op(insrt+1..nops(pi),pi)] ;
        end if;
    end do:
    return pi ;
end proc:
A119441 := proc(n,k)
    local pi,a,p ;
    pi := ASPrts(n)[k] ;
    a := 1 ;
    for p in pi do
        a := a*combinat[numbpart](p) ;
    end do:
    a ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A119441(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013

A296150 Triangle whose n-th row is the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 2, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 3, 2, 1, 1, 1, 1, 7, 2, 2, 1, 8, 3, 1, 1, 4, 2, 5, 1, 9, 2, 1, 1, 1, 3, 3, 6, 1, 2, 2, 2, 4, 1, 1, 10, 3, 2, 1, 11, 1, 1, 1, 1, 1, 5, 2, 7, 1, 4, 3, 2, 2, 1, 1, 12, 8, 1, 6, 2, 3, 1, 1, 1, 13, 4

Offset: 1

Gus Wiseman, Feb 05 2018

Same as A112798 with rows reversed. Row lengths are A001222. Rows sums are A056239.

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

			Sequence of partitions begins: (), (1), (2), (11), (3), (21), (4), (111), (22), (31), (5), (211), (6), (41), (32), (1111), (7), (221).

Robert Israel, Table of n, a(n) for n = 1..10002 (rows 1 to 3272, flattened)

Cf. A000041, A000720, A001222, A056239, A063834, A112798, A196545, A215366, A289501, A299200, A299201, A299202, A299203.

f := n -> op(map(numtheory:-pi, sort(map(`$`@op, ifactors(n)[2]), `>`))):
map(f, [$1..100]); # Robert Israel, Feb 09 2018

Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,50}]

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
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).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

Original entry on oeis.org

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004
Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018
Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 03 2018

nonn

A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.

			Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.

with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 07 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[PrimeOmega/@primeMS[n]],{n,100}]

a(n,f=factor(n))=sum(i=1,#f~, bigomega(primepi(f[i,1]))*f[i,2]) \\ Charles R Greathouse IV, Nov 10 2021

A276024 Number of positive subset sums of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 14, 27, 47, 81, 130, 210, 319, 492, 718, 1063, 1512, 2178, 3012, 4237, 5765, 7930, 10613, 14364, 18936, 25259, 32938, 43302, 55862, 72694, 92797, 119499, 151468, 193052, 242748, 307135, 383315, 481301, 597252, 744199, 918030, 1137607, 1395101, 1718237, 2098096, 2569047, 3121825, 3805722

Offset: 1

Gus Wiseman, Aug 16 2016

nonn

For a multiset p of positive integers summing to n, a pair (t,p) is defined to be a positive subset sum if there exists a nonempty submultiset of p summing to t. Positive integers with positive subset sums form a multiorder. This sequence is dominated by A122768 (submultisets of integer partitions of n).

			The a(4)=14 positive subset sums are: {(4,4), (1,31), (3,31), (4,31), (2,22), (4,22), (1,211), (2,211), (3,211), (4,211), (1,1111), (2,1111), (3,1111), (4,1111)}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150
Konstantinos Koiliaris and Chao Xu, A Faster Pseudopolynomial Time Algorithm for Subset Sum, arXiv:1507.02318 [cs.DS], 2015-2016.
Gus Wiseman, Comcategories and Multiorders (pdf version)

Cf. A122768, A063834, A262671.

sums[ptn_?OrderedQ]:=sums[ptn]=If[Length[ptn]===1,ptn,Module[{pri,sms},
pri=Union[Table[Delete[ptn,i],{i,Length[ptn]}]];
sms=Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@pri;
Union@@sms
]];
Table[Total[Length[sums[Sort[#]]]&/@IntegerPartitions[n]],{n,1,25}]
(* Second program: *)
b[n_, i_, s_] := b[n, i, s] = If[n == 0, Length[s], If[i < 1, 0, b[n, i - 1, s] + b[n - i, Min[n - i, i], {#, # + i}& /@ s // Flatten // Union]]];
a[n_] := b[n, n, {0}] - PartitionsP[n];
Array[a, 45] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz in A304792 *)

# uses A304792_T
from sympy import npartitions
def A276024(n): return A304792_T(n,n,(0,),1) - npartitions(n) # Chai Wah Wu, Sep 25 2023

A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909

Offset: 0

Vaclav Kotesovec, Mar 28 2016

nonn

The number of ways a number can be partitioned into not necessarily distinct parts and then each part is partitioned into distinct parts. Also a(n) > A089259(n) for n>5. - Gus Wiseman, Apr 10 2016
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers with weakly decreasing multiplicities. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 7 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{2},{1,1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{1},{2},{1,1}}
                                   {{2},{3},{1,1}}
                                   {{1},{2},{3},{4}}
The weakly normal non-strict version is A055887.
The non-strict version is A063834.
The weakly normal version is A304969.
(End)

			a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.
The unordered version is A089259, non-strict A001970 (row-sums of A061260).
For compositions instead of partitions we have A304969, non-strict A055887.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by sum and length.

nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

From Vaclav Kotesovec, Mar 28 2016: (Start)
a(n) ~ c * n^2 * 2^(n/3), where
c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0
c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1
c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2
(End)

A300061 Heinz numbers of integer partitions of even numbers.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 30, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 66, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 102, 107, 108, 111, 112, 113, 115, 116, 117, 118, 120

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

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

			75 is the Heinz number of (3,3,2), which has even weight, so 75 belongs to the sequence.
Sequence of even-weight partitions begins: () (2) (1,1) (4) (2,2) (3,1) (2,1,1) (6) (1,1,1,1) (8) (4,2) (5,1) (3,3) (2,2,2) (4,1,1).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Complement of A300063.

Cf. A000041, A000720, A001222, A056239, A063834, A100118, A112798, A122111, A215366, A296150, A299202, A299757, A300056, A300060, A304662.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 0, a(n-1)) while add(numtheory[pi]
      (i[1])*i[2], i=ifactors(k)[2])::odd do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 22 2018

Select[Range[200],EvenQ[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]]&]

A281113 Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 15, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 28, 3, 3, 3, 32, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 58, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 66, 3, 12, 1, 9, 3, 12, 1, 84, 1, 3, 9, 9, 3, 12, 1, 58, 15, 3

Offset: 2

Gus Wiseman, Jan 14 2017

nonn

A postpositive number is a positive integer other than 1. A postpositive factorization of n is a finite orderless sequence of postpositive numbers whose product is n.

			The a(20)=9 twice-factorizations are: ((20)), ((2*10)), ((4*5)), ((2*2*5)), ((2)*(10)), ((2)*(2*5)), ((4)*(5)), ((2*2)*(5)), ((2)*(2)*(5)).
Twice-factorizations of 32 organized by composite:
((2)(2)(2)(2)(2)) ((2)(2)(2)(2 2)) ((2)(2)(2 2 2)) ((2)(2 2)(2 2)) ((2)(2 2 2 2)) ((2 2)(2 2 2)) ((2 2 2 2 2))
((2)(2)(2)(4))    ((2)(2)(2 4))    ((2)(2 2)(4))   ((2)(4)(2 2))   ((2)(2 2 4))   ((2 2)(2 4))   ((4)(2 2 2))  ((2 2 2 4))
((2)(2)(8))       ((2)(2 8))       ((2 2)(8))      ((2 2 8))
((2)(4)(4))       ((2)(4 4))       ((4)(2 4))      ((2 4 4))
((2)(16))         ((2 16))
((4)(8))          ((4 8))
((32)).
Twice-factorizations of 32 organized by domain:
((2)(2)(2)(2)(2))
((2)(2)(2)(2 2)) ((2)(2)(2)(4))
((2)(2)(2 2 2))  ((2)(2)(2 4)) ((2)(2)(8))
((2)(2 2)(2 2))  ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(4)(4))
((2)(2 2 2 2))   ((2)(2 2 4))  ((2)(2 8))    ((2)(4 4))   ((2)(16))
((2 2)(2 2 2))   ((2 2)(2 4))  ((2 2)(8))    ((4)(2 2 2)) ((4)(2 4)) ((4)(8))
((2 2 2 2 2))    ((2 2 2 4))   ((2 2 8))     ((2 4 4))    ((2 16))   ((4 8)) ((32)).

Michael De Vlieger, Table of n, a(n) for n = 2..30030
Michael De Vlieger, Indices of records in A281113.

Cf. A001055(n) = number of factorizations of n, A050336(n) = number of orderless twice-factorizations of n, A162247(n) = factors of factorizations of n, A063834(n) = a(p^(n-1)), A007716, A269134, A281116.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twicefacs[n_]:=Join@@Tuples/@Map[postfacs,postfacs[n],{2}];
Table[Length[twicefacs[n]],{n,2,24}]

A196545 Number of weakly ordered plane trees with n leaves.

Original entry on oeis.org

1, 1, 2, 5, 12, 34, 92, 277, 806, 2500, 7578, 24198, 75370, 243800, 776494, 2545777, 8223352, 27221690, 88984144, 296856400, 979829772, 3287985078, 10934749788, 36912408342, 123519937044, 418650924886, 1408867195252, 4794243983204, 16205061000480

Offset: 1

Gus Wiseman, Oct 03 2011

nonn

A weakly ordered plane tree (p-tree) of weight n > 1 is a sequence (t_1, t_2, ..., t_k) where k > 1 and for some integer partition y of length k and sum n, the term t_i is a p-tree of weight y_i, for 1 <= i <= k. For n = 1, the only p-tree is a single node.

This definition precludes nodes with only one branch, and non-leaf nodes have weight 0. If the above is changed so that k >= 1 and y is partition of n-1, we get the trees counted by A093637. Binary p-trees are counted by A000992.

			Let o denote a single node. The 12 p-trees of weight 5 are: ((((oo)o)o)o), (((ooo)o)o), (((oo)(oo))o), (((oo)oo)o), ((oooo)o), (((oo)o)(oo)), ((ooo)(oo)), (((oo)o)oo), ((ooo)oo), ((oo)(oo)o), ((oo)ooo), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
David Callan, The weakly ordered plane trees of size up to 4
Gus Wiseman, All 277 weakly ordered plane trees of weight 8

Cf. A000041, A063834, A093637, A000992.

b:= proc(n, i) option remember;
      `if`(i>n, 0, a(i)*`if`(i=n, 1, b(n-i, i)))+`if`(i>1, b(n, i-1), 0)
    end:
a:= proc(n) option remember;
      `if`(n=1, 1, add(a(k)*b(n-k, min(n-k, k)), k=1..n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 06 2012

PTNum[n_] := PTNum[n] =
    If[n === 1, 1, Plus @@ Function[y,
    Times @@ PTNum /@ y] /@ Rest[Partitions[n]]]; Array[PTNum, 20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[i>n, 0, a[i]*If[i == n, 1, b[n-i, i]]] + If[i>1, b[n, i-1], 0];
a[n_] := a[n] = If[n == 1, 1, Sum[a[k]*b[n-k, Min[n-k, k] ], {k, 1, n-1}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); v[1] = 1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); v} \\ Andrew Howroyd, Aug 26 2018

@cached_function
def A196545(n):
    if n == 1: return 1
    return sum(prod(A196545(t) for t in p) for p in Partitions(n,min_length=2))
# D. S. McNeil, Oct 03 2011

a(1) = 1; for n > 1, a(n) = Sum_{y} Product_{i} a(y_i) where the sum is over all partitions of n with at least two parts.

The generating function is characterized by the formal equation 1 + 2*S(x) = x + 1/P(x) where S(x) = Sum_{n>0} a(n)*x^n and P(x) = Product_{n>0} (1 - a(n)*x^n) are the formal infinite series and formal infinite product with a(n) as coefficients and roots respectively.

cp's OEIS Frontend

A119441 Distribution of A063834 in Abramowitz and Stegun order.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A296150 Triangle whose n-th row is the integer partition with Heinz number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A276024 Number of positive subset sums of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

Views

Author

Keywords

Comments