A320924 -id:A320924 - cp's OEIS frontend

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, 99133, 120770, 146785, 177970, 215308, 259891, 313065, 376326, 451501

Offset: 0

N. J. A. Sloane

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004
With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
Also, number of partitions of n into parts when there are two kinds of parts of size one.
Also number of graphical forest partitions of 2n+2.
a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004
Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005
Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005
Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008
Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008
Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009
a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011
Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012
With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013
a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016
a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018
a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example,  D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018
With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018
Also a(n) is the number of partitions of 2n+1 with exactly one odd part.
Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019
In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021
a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024
a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 45*x^7 + 67*x^8 + ...
From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 5 consider the partitions of n+1:
--------------------------------------
.                         Number
Partitions of 6           of 1's
--------------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 0
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
35-16 =                     19
.
The difference between the sum of the first column and the sum of the second column of the set of partitions of 6 is 35 - 16 = 19 and equals the number of 1's in all partitions of 6, so the 6th term of this sequence is a(5) = 19.
(End)
From _Gus Wiseman_, Oct 26 2018: (Start)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose greatest part is > n:
  (2)  (4)   (6)    (8)     (A)      (C)
       (31)  (42)   (53)    (64)     (75)
             (51)   (62)    (73)     (84)
             (411)  (71)    (82)     (93)
                    (521)   (91)     (A2)
                    (611)   (622)    (B1)
                    (5111)  (631)    (732)
                            (721)    (741)
                            (811)    (822)
                            (6211)   (831)
                            (7111)   (921)
                            (61111)  (A11)
                                     (7221)
                                     (7311)
                                     (8211)
                                     (9111)
                                     (72111)
                                     (81111)
                                     (711111)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose number of parts is > n:
  (11)  (211)   (2211)    (22211)     (222211)      (2222211)
        (1111)  (3111)    (32111)     (322111)      (3222111)
                (21111)   (41111)     (331111)      (3321111)
                (111111)  (221111)    (421111)      (4221111)
                          (311111)    (511111)      (4311111)
                          (2111111)   (2221111)     (5211111)
                          (11111111)  (3211111)     (6111111)
                                      (4111111)     (22221111)
                                      (22111111)    (32211111)
                                      (31111111)    (33111111)
                                      (211111111)   (42111111)
                                      (1111111111)  (51111111)
                                                    (222111111)
                                                    (321111111)
                                                    (411111111)
                                                    (2211111111)
                                                    (3111111111)
                                                    (21111111111)
                                                    (111111111111)
(End)
From _Joerg Arndt_, Jan 01 2024: (Start)
The a(5) = 19 multiset partitions of the multiset {1^5, 2^1} are:
   1:  {{1, 1, 1, 1, 1, 2}}
   2:  {{1, 1, 1, 1, 1}, {2}}
   3:  {{1, 1, 1, 1, 2}, {1}}
   4:  {{1, 1, 1, 1}, {1, 2}}
   5:  {{1, 1, 1, 1}, {1}, {2}}
   6:  {{1, 1, 1, 2}, {1, 1}}
   7:  {{1, 1, 1, 2}, {1}, {1}}
   8:  {{1, 1, 1}, {1, 1, 2}}
   9:  {{1, 1, 1}, {1, 1}, {2}}
  10:  {{1, 1, 1}, {1, 2}, {1}}
  11:  {{1, 1, 1}, {1}, {1}, {2}}
  12:  {{1, 1, 2}, {1, 1}, {1}}
  13:  {{1, 1, 2}, {1}, {1}, {1}}
  14:  {{1, 1}, {1, 1}, {1, 2}}
  15:  {{1, 1}, {1, 1}, {1}, {2}}
  16:  {{1, 1}, {1, 2}, {1}, {1}}
  17:  {{1, 1}, {1}, {1}, {1}, {2}}
  18:  {{1, 2}, {1}, {1}, {1}, {1}}
  19:  {{1}, {1}, {1}, {1}, {1}, {2}}
(End)

H. Gupta, An asymptotic formula in partitions. J. Indian Math. Soc., (N. S.) 10 (1946), 73-76.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
Stanley, R. P., Exercise 1.26 in Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, p. 59, 1999.

T. D. Noe, Table of n, a(n) for n = 0..1000
P. A. Baikov and S. V. Mikhailov, The {beta}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O(alpha_s^4), J. High Energy Phys. 09 (2022) Art. No. 185. See also arXiv:2206.14063 [hep-ph], 2022.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See p. 15.
L. Bracci and L. E. Picasso, A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics, The European Physical Journal Plus, 127 (2012), Article 119. - From _N. J. A. Sloane_, Dec 31 2012
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D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, preprint.
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Petros Hadjicostas, Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer, Journal of Integer Sequences, 20 (2017), Article #17.8.5.
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Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
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M. D. Hirschhorn, The number of different parts in the partitions of n, Fib. Quart., 52 (2014), 10-15. See p. 11. - _N. J. A. Sloane_, Mar 25 2014
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A diagonal of A066633.
Also second column of A126442. - George Beck, May 07 2011
Row sums of triangle A092905.
Also row sums of triangle A261555. - Omar E. Pol, Sep 14 2016
Also row sums of triangle A278427. - John P. McSorley, Nov 25 2016
Column k=2 of A292508.
Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.
Cf. A000569, A025065, A036469, A096373, A147878, A209816, A320891, A320924.

List([0..45],n->Sum([0..n],k->NrPartitions(k))); # Muniru A Asiru, Jul 25 2018

a000070 = p a028310_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

with(combinat): a:=n->add(numbpart(j),j=0..n): seq(a(n), n=0..44); # Zerinvary Lajos, Aug 26 2008

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Table[ Count[ Flatten@ IntegerPartitions@ n, 1], {n, 45}] (* Robert G. Wilson v, Aug 06 2008 *)
Join[{1}, Accumulate[PartitionsP[Range[50]]]+1] (* _Harvey P. Dale, Mar 12 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (1 - x) / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 09 2013 *)
Accumulate[PartitionsP[Range[0,49]]] (* George Beck, Oct 23 2014; typo fixed by Virgile Andreani, Jul 10 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(m=1, n, 1 - x^m, 1 + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Nov 08 2002 */

x='x+O('x^66); Vec(1/((1-x)*eta(x))) /* Joerg Arndt, May 15 2011 */

a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016

from itertools import accumulate
def A000070iter(n):
    L = [0]*n; L[0] = 1
    def numpart(n):
        S = 0; J = n-1; k = 2
        while 0 <= J:
            T = L[J]
            S = S+T if (k//2)%2 else S-T
            J -= k  if (k)%2 else k//2
            k += 1
        return S
    for j in range(1, n): L[j] = numpart(j)
    return accumulate(L)
print(list(A000070iter(100))) # Peter Luschny, Aug 30 2019

# Using function A365676Row. Compare also A365675.
from itertools import accumulate
def A000070List(size: int) -> list[int]:
    return [sum(accumulate(reversed(A365676Row(n)))) for n in range(size)]
print(A000070List(45))  # Peter Luschny, Sep 16 2023

def A000070_list(leng):
    p = [number_of_partitions(n) for n in range(leng)]
    return [add(p[:k+1]) for k in range(leng)]
A000070_list(45) # Peter Luschny, Sep 15 2014

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
log(a(n)) ~ -3.3959 + 2.44613*sqrt(n). - Robert G. Wilson v, Jan 11 2002
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1 - x))*Product_{m >= 1} 1/(1 - x^m).
a(n) seems to have the same parity as A027349(n+1). Comment from James Sellers, Mar 08 2006: that is true.
a(n) = A000041(2n+1) - A110618(2n+1) = A000041(2n+2) - A110618(2n+2). - Henry Bottomley, Aug 01 2005
Row sums of triangle A133735. - Gary W. Adamson, Sep 22 2007
a(n) = A092269(n+1) - A195820(n+1). - Omar E. Pol, Oct 20 2011
a(n) = A181187(n+1,1) - A181187(n+1,2). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 23 2013: (Start)
Gupta gives the asymptotic result a(n-1) ~ sqrt(6/Pi^2)* sqrt(n)*p(n), where p(n) is the partition function A000041(n).
Let P(2,n) denote the set of partitions of n into parts k >= 2.
a(n-2) = Sum_{parts k in all partitions in P(2,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, leads to the asymptotic result
a(n-2) ~ (6/Pi^2)*n*(p(n) - p(n-1)) = (6/Pi^2)*A138880(n) as n -> infinity. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n)) + (73*Pi^2 - 1584)/(6912*n)). - Vaclav Kotesovec, Oct 26 2016
a(n) = A024786(n+2) + A024786(n+1). - Vaclav Kotesovec, Nov 05 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) = A025065(2n). - Gus Wiseman, Oct 26 2018
a(n - 1) = A000041(2n) - A209816(n). - Gus Wiseman, Oct 26 2018

A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Gus Wiseman, Feb 15 2022

nonn

First differs from A130091 (Wilf partitions) in having 216.
See A239455 for the definition of Look-and-Say partitions.
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.

			The terms together with their prime indices begin:
      1: ()            20: (3,1,1)         47: (15)
      2: (1)           23: (9)             48: (2,1,1,1,1)
      3: (2)           24: (2,1,1,1)       49: (4,4)
      4: (1,1)         25: (3,3)           50: (3,3,1)
      5: (3)           27: (2,2,2)         52: (6,1,1)
      7: (4)           28: (4,1,1)         53: (16)
      8: (1,1,1)       29: (10)            54: (2,2,2,1)
      9: (2,2)         31: (11)            56: (4,1,1,1)
     11: (5)           32: (1,1,1,1,1)     59: (17)
     12: (2,1,1)       37: (12)            61: (18)
     13: (6)           40: (3,1,1,1)       63: (4,2,2)
     16: (1,1,1,1)     41: (13)            64: (1,1,1,1,1,1)
     17: (7)           43: (14)            67: (19)
     18: (2,2,1)       44: (5,1,1)         68: (7,1,1)
     19: (8)           45: (3,2,2)         71: (20)
For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).

The Wilf case (distinct multiplicities) is A130091, counted by A098859.
The complement of the Wilf case is A130092, counted by A336866.
These partitions appear to be counted by A239455.
A variant for runs is A351201, counted by A351203 (complement A351204).
The complement is A351295, counted by A351293.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of run-lengths in binary expansion, for all runs A297770.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]!={}&]

Name edited by Gus Wiseman, Aug 13 2025

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

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.

			The terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

A209816 Number of partitions of 2n in which every part is

Original entry on oeis.org

1, 3, 7, 15, 30, 58, 105, 186, 318, 530, 863, 1380, 2164, 3345, 5096, 7665, 11395, 16765, 24418, 35251, 50460, 71669, 101050, 141510, 196888, 272293, 374423, 512081, 696760, 943442, 1271527, 1706159, 2279700, 3033772, 4021695, 5311627, 6990367, 9168321

Offset: 1

Clark Kimberling, Mar 13 2012

nonn

Also, the number of partitions of 3n in which n is the maximal part.
Also, the number of partitions of 3n into n parts. - Seiichi Manyama, May 07 2018
Also the number of multigraphical partitions of 2n, i.e., integer partitions that comprise the multiset of vertex-degrees of some multigraph. - Gus Wiseman, Oct 24 2018
Also number of partitions of 2n with at most n parts. Conjugate partitions map one to one to partitions of 2*n with each part <= n. - Wolfdieter Lang, May 21 2019

			The 7 partitions of 6 with parts <4 are as follows:
3+3, 3+2+1, 3+1+1+1
2+2+2, 2+2+1+1, 2+1+1+1+1
1+1+1+1+1+1.
Matching partitions of 2 into rationals as described:
1 + 1
1 + 3/3 + 1/3
1 + 1/3 + 1/3 + 1/3
2/3 + 2/3 + 2/3
2/3 + 2/3 + 1/3 + 1/3
2/3 + 1/3 + 1/3 + 1/3 + 1/3
1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.
From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of 3n into n parts
--+-------------------------------------------------
1 | 3;
2 | 5+1, 4+2, 3+3;
3 | 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3; (End)
From _Gus Wiseman_, Oct 24 2018: (Start)
The a(1) = 1 through a(4) = 15 partitions:
  (11)  (22)    (33)      (44)
        (211)   (222)     (332)
        (1111)  (321)     (422)
                (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Multigraphs realizing each of the a(4) = 15 multigraphical partitions of 8.
Gus Wiseman, Multigraphs realizing each of the a(5) = 30 multigraphical partitions of 10.

Cf. A000041, A000070, A000569, A008284, A025065, A079122, A096373, A147878, A209815, A320911, A320921, A320924.

a209816 n = p [1..n] (2*n) where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 14 2013

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012

f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n &]]; Table[f[n], {n, 1, 30}] (* A209816 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,2*n}],{n,1,20}] (* Vaclav Kotesovec, May 25 2015 *)
Table[Length@IntegerPartitions[3n, {n}], {n, 25}] (* Vladimir Reshetnikov, Jul 24 2016 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = A000041(2*n)-A000070(n-1). - Matthew Vandermast, Jul 16 2012

a(n) = Sum_{k=1..n} A008284(2*n, k) = A000041(2*n) - A000070(n-1), for n >= 1. - Wolfdieter Lang, May 21 2019

More terms from Alois P. Heinz, Jul 09 2012

A004250 Number of partitions of n into 3 or more parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, 127, 168, 222, 288, 375, 480, 616, 781, 990, 1243, 1562, 1945, 2422, 2996, 3703, 4550, 5588, 6826, 8332, 10126, 12292, 14865, 17958, 21618, 25995, 31165, 37317, 44562

Offset: 1

N. J. A. Sloane

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014

Also the number of graphical partitions of 2n into n parts. - Gus Wiseman, Jan 08 2021

			a(6)=7 because there are three partitions of n=6 with i=3 parts: [4, 1, 1], [3, 2, 1], [2, 2, 2] and two partitions with i=4 parts: [3, 1, 1, 1], [2, 2, 1, 1] and one partition with i=5 parts: [2, 1, 1, 1, 1] and one partition with i=6 parts: [1, 1, 1, 1, 1, 1].
From _Gus Wiseman_, Jan 18 2021: (Start)
The a(3) = 1 through a(7) = 11 graphical partitions of 2n into n parts:
  (222)  (2222)  (22222)  (222222)  (2222222)
         (3221)  (32221)  (322221)  (3222221)
                 (33211)  (332211)  (3322211)
                 (42211)  (333111)  (3332111)
                          (422211)  (4222211)
                          (432111)  (4322111)
                          (522111)  (4331111)
                                    (4421111)
                                    (5222111)
                                    (5321111)
                                    (6221111)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
N. Metropolis and P. R. Stein, The enumeration of graphical partitions, Europ. J. Combin., 1 (1980), 139-1532.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics. Spider Graph
Wikipedia, Starlike tree
Index entries for sequences related to graphical partitions

Cf. A004251, A029889, A035300, A095268, A058984.
Rightmost column of A259873.
Central column of A339659.
A000041 counts partitions of 2n into n parts, ranked by A340387.
A000569 counts graphical partitions, ranked by A320922.
A008284 counts partitions by sum and length.
A027187 counts partitions of even length.
A309356 ranks simple covering graphs.
The following count vertex-degree partitions and give their Heinz numbers:
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339617 counts non-graphical partitions of 2n (A339618).
- A339656 counts loop-graphical partitions (A339658).
Cf. A000070, A000219, A339560, A339561, A339661.
Partial sums of A117995.

with(combinat);
for i from 1 to 15 do pik(i,3) od;
pik:= proc(n::integer, k::integer)
# Thomas Wieder, Jan 30 2007
local i, Liste, Result;
if k > n or n < 0 or k < 1 then
return fail
end if;
Result := 0;
for i from k to n do
Liste:= PartitionList(n,i);
#print(Liste);
Result := Result + nops(Liste);
end do;
return Result;
end proc;
PartitionList := proc (n, k)
# Authors: Herbert S. Wilf and Joanna Nordlicht. Source: Lecture Notes
# "East Side West Side,..." University of Pennsylvania, USA, 2002.
# Available at: http://www.cis.upenn.edu/~wilf/lecnotes.html
# Calculates the partition of n into k parts.
# E.g. PartitionList(5,2) --> [[4, 1], [3, 2]].
local East, West;
if n < 1 or k < 1 or n < k then
RETURN([])
elif n = 1 then
RETURN([[1]])
else if n < 2 or k < 2 or n < k then
West := []
else
West := map(proc (x) options operator, arrow;
[op(x), 1] end proc,PartitionList(n-1,k-1)) end if;
if k <= n-k then
East := map(proc (y) options operator, arrow;
map(proc (x) options operator, arrow; x+1 end proc,y) end proc,PartitionList(n-k,k))
else East := [] end if;
RETURN([op(West), op(East)])
end if;
end proc;
#  Thomas Wieder, Feb 01 2007
ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..41); # Zerinvary Lajos, Mar 25 2008
B:=[S,{S = Set(Sequence(Z,1 <= card),card >=3)},unlabelled]: seq(combstruct[count](B, size=n), n=1..41); # Zerinvary Lajos, Mar 21 2009

Length /@ Table[Select[Partitions[n], Length[#] > 2 &], {n, 20}] (* Eric W. Weisstein, May 16 2007 *)
Table[Count[Length /@ Partitions[n], ?(# > 2 &)], {n, 20}] (* _Eric W. Weisstein, May 16 2017 *)
Table[PartitionsP[n] - Floor[n/2] - 1, {n, 20}] (* Eric W. Weisstein, May 16 2017 *)
Length /@ Table[IntegerPartitions[n, {3, n}], {n, 20}] (* Eric W. Weisstein, May 16 2017 *)

a(n) = numbpart(n) - (n+2)\2; /* Joerg Arndt, Apr 03 2013 */

G.f.: Sum_{n>=0} (q^n / Product_{k=1..n+3} (1 - q^k)). -  N. J. A. Sloane
a(n) = A000041(n) - floor((n+2)/2) = A000041(n)-A004526(n+2) = A058984(n)-1. - Vladeta Jovovic, Jun 18 2003
Let P(n,i) denote the number of partitions of n into i parts. Then a(n) = Sum_{i=3..n} P(n,i). - Thomas Wieder, Feb 01 2007
a(n) = A259873(n,n). - Gus Wiseman, Jan 08 2021

Definition corrected by Thomas Wieder, Feb 01 2007 and by Eric W. Weisstein, May 16 2007

A320922 Heinz numbers of graphical partitions.

Original entry on oeis.org

1, 4, 12, 16, 27, 36, 40, 48, 64, 81, 90, 108, 112, 120, 144, 160, 192, 225, 243, 252, 256, 270, 300, 324, 336, 352, 360, 400, 432, 448, 480, 567, 576, 625, 630, 640, 675, 729, 750, 756, 768, 792, 810, 832, 840, 900, 972, 1000, 1008, 1024, 1056, 1080, 1120

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

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

An integer partition is graphical if it comprises the vertex-degrees of some simple graph.

			The sequence of all graphical partitions begins: (), (11), (211), (1111), (222), (2211), (3111), (21111), (111111), (2222), (3221), (22211), (41111), (32111), (221111), (311111), (2111111), (3322), (22222), (42211).

Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911, A320923, A320924.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],UnsameQ@@#&]!={}&]

A086543 Number of partitions of n with at least one odd part.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 15, 17, 30, 35, 56, 66, 101, 120, 176, 209, 297, 355, 490, 585, 792, 946, 1255, 1498, 1958, 2335, 3010, 3583, 4565, 5428, 6842, 8118, 10143, 12013, 14883, 17592, 21637, 25525, 31185, 36711, 44583, 52382, 63261, 74173, 89134, 104303, 124754, 145698, 173525, 202268

Offset: 0

Vladeta Jovovic, Sep 10 2003

nonn

From Gus Wiseman, Oct 12 2023: (Start)
Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)
            (21)   (31)    (32)     (42)      (43)
            (111)  (1111)  (41)     (51)      (52)
                           (221)    (222)     (61)
                           (311)    (411)     (322)
                           (2111)   (2211)    (331)
                           (11111)  (21111)   (421)
                                    (111111)  (511)
                                              (2221)
                                              (3211)
                                              (4111)
                                              (22111)
                                              (31111)
                                              (211111)
                                              (1111111)
Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:
  (1)  (11)  (3)    (31)    (5)      (33)      (7)
             (21)   (211)   (32)     (51)      (43)
             (111)  (1111)  (41)     (321)     (52)
                            (221)    (411)     (61)
                            (311)    (2211)    (322)
                            (2111)   (3111)    (331)
                            (11111)  (21111)   (421)
                                     (111111)  (511)
                                               (2221)
                                               (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

			a(4)=3 because we have [3,1],[2,1,1] and [1,1,1] ([4] and [2,2] do not qualify).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A038348, A047967.
The complement is counted by A035363, ranks A344415.
These partitions have ranks A366322.
A025065 counts partitions with sum <= twice length, ranks A344296.
A110618 counts partitions with sum >= twice maximum, ranks A344291.
Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.

g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # Emeric Deutsch, Mar 30 2006

nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)
Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* Gus Wiseman, Oct 12 2023 *)

x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ Joerg Arndt, May 04 2013

A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).
G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - Joerg Arndt, May 04 2013

A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 5, 8, 8, 13, 17, 22, 28, 39, 48, 62, 81, 101, 127, 167, 202, 253, 318, 395, 486, 608, 736, 906, 1113, 1353, 1637, 2011, 2409, 2922, 3510, 4227, 5060, 6089, 7242, 8661, 10306, 12251, 14503, 17236, 20345, 24045, 28334, 33374, 39223, 46076

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

Naturally, such a partition must have an even number of parts. Its multiplicities form a graphical partition (A000569, A320922), and vice versa.

			The a(3) = 1 through a(11) = 13 partitions (A = 10):
  (21)  (31)  (32)  (42)  (43)    (53)    (54)    (64)    (65)
              (41)  (51)  (52)    (62)    (63)    (73)    (74)
                          (61)    (71)    (72)    (82)    (83)
                          (3211)  (3221)  (81)    (91)    (92)
                                  (4211)  (3321)  (4321)  (A1)
                                          (4221)  (5221)  (4322)
                                          (4311)  (5311)  (4331)
                                          (5211)  (6211)  (4421)
                                                          (5321)
                                                          (5411)
                                                          (6221)
                                                          (6311)
                                                          (7211)
For example, the partition y = (4,3,3,2,1,1) can be partitioned into a set of edges in two ways:
  {{1,2},{1,3},{3,4}}
  {{1,3},{1,4},{2,3}},
so y is counted under a(14).

Eric Weisstein's World of Mathematics, Graphical partition.

A338916 allows equal pairs (x,x).
A339559 counts the complement in even-length partitions.
A339561 gives the Heinz numbers of these partitions.
A339619 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
A339659 counts graphical partitions of 2n into k parts.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A320893, A338899, A338903, A339564.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],strs[Times@@Prime/@#]!={}&]],{n,0,15}]

A027187(n) = a(n) + A339559(n).

More terms from Jinyuan Wang, Feb 14 2025

A339561 Products of distinct squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166

Offset: 1

Gus Wiseman, Dec 13 2020

nonn

First differs from A320911 in lacking 36.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      1: {}        55: {3,5}         91: {4,6}
      6: {1,2}     57: {2,8}         93: {2,11}
     10: {1,3}     58: {1,10}        94: {1,15}
     14: {1,4}     60: {1,1,2,3}     95: {3,8}
     15: {2,3}     62: {1,11}       106: {1,16}
     21: {2,4}     65: {3,6}        111: {2,12}
     22: {1,5}     69: {2,9}        115: {3,9}
     26: {1,6}     74: {1,12}       118: {1,17}
     33: {2,5}     77: {4,5}        119: {4,7}
     34: {1,7}     82: {1,13}       122: {1,18}
     35: {3,4}     84: {1,1,2,4}    123: {2,13}
     38: {1,8}     85: {3,7}        126: {1,2,2,4}
     39: {2,6}     86: {1,14}       129: {2,14}
     46: {1,9}     87: {2,10}       132: {1,1,2,5}
     51: {2,7}     90: {1,2,2,3}    133: {4,8}
For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
  (6*10*15*77)
  (6*10*21*55)
  (6*10*33*35)
  (6*14*15*55)
  (6*15*22*35)
  (10*14*15*33)
  (10*15*21*22),
so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A309356 is a kind of universal embedding.
A320894 is the complement in A028260.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A339560 counts the partitions with these Heinz numbers.
A339661 has nonzero terms at these positions.
A001358 lists semiprimes, with squarefree case A006881.
A005117 lists squarefree numbers.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqs[#]!={}&]

A339561 \/ A320894 = A028260.

cp's OEIS Frontend

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

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A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A209816 Number of partitions of 2n in which every part is

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A004250 Number of partitions of n into 3 or more parts.

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A320922 Heinz numbers of graphical partitions.

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