A320894 -id:A320894 - cp's OEIS frontend

A035363 Number of partitions of n into even parts.

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- 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).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose prime multiplicities do not exceed x/2.

			360 is in the sequence because it can be factored into squarefree semiprimes as (6*6*10).
4620 is in the sequence, and can be factored into squarefree semiprimes in 6 ways: (6*10*77), (6*14*55), (6*22*35), (10*14*33), (10*21*22), (14*15*22).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001222, A001358, A005117, A006881, A007717, A028260, A320655, A320656, A320891, A320892, A320893, A320894, A320912.

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

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

Original entry on oeis.org

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

A320912 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into distinct semiprimes.

Original entry on oeis.org

1, 4, 6, 9, 10, 14, 15, 21, 22, 24, 25, 26, 33, 34, 35, 36, 38, 39, 40, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 100, 104, 106, 111, 115, 118, 119, 121, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.

			9000 is in the sequence and can be factored in either of two ways: (4*6*15*25) or (4*9*10*25).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001222, A001358, A007717, A028260, A320655, A320892, A320893, A320894, A320911.

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

A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.

Original entry on oeis.org

4, 9, 16, 24, 25, 40, 49, 54, 56, 64, 81, 88, 96, 104, 121, 135, 136, 144, 152, 160, 169, 184, 189, 224, 232, 240, 248, 250, 256, 289, 296, 297, 324, 328, 336, 344, 351, 352, 361, 375, 376, 384, 400, 416, 424, 459, 472, 486, 488, 513, 528, 529, 536, 544, 560

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A squarefree semiprime (A006881) is a product of any two distinct primes.

Also numbers with an even number x of prime factors, whose greatest prime multiplicity exceeds x/2.

			A complete list of all factorizations of 24 is:
  (2*2*2*3),
  (2*2*6), (2*3*4),
  (2*12), (3*8), (4*6),
  (24).
All of these contain at least one number that is not a squarefree semiprime, so 24 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320892, A320893, A320894.

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

A338900 Difference between the two prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 5, 3, 6, 1, 7, 4, 8, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 15, 23, 16, 9, 2, 24, 17, 25, 6, 10, 26, 3, 18, 27, 11, 7, 28, 19, 1, 29, 12, 20, 2, 21, 4

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

Is this sequence an anti-run, i.e., are there no adjacent equal parts? I have verified this conjecture up to n = 10^6. - Gus Wiseman, Nov 18 2020

A176506 is the not necessarily squarefree version.
A338899 has row-differences equal to this sequence.
A338901 gives positions of first appearances.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes.
A002100 and A338903 count partitions using squarefree semiprimes.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A065516 gives first differences of semiprimes.
A166237 gives first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A000040, A056239, A112798, A167171, A320656, A320891, A320894, A320911, A338898, A338905, A338908.

-Subtract@@PrimePi/@First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]

If the n-th squarefree semiprime is prime(x) * prime(y) with x < y, then a(n) = y - x.

a(n) = A270652(n) - A270650(n).

A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 102, 342, 1213, 4361, 16016, 59348, 222117, 836315, 3166852, 12042620, 45967479, 176005709, 675759564, 2600672458, 10029832754, 38753710486, 149990133774, 581393603996, 2256710139346, 8770547818956, 34125389919850, 132919443189544, 518232001761434, 2022337118015338, 7898574056034636, 30873421455729728

Offset: 0

N. J. A. Sloane

In other words, a(n) is the number of graphic sequences of length n, where a graphic sequence is a sequence of numbers which can be the degree sequence of some graph.

In the article by A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, "Parallel enumeration of degree sequences of simple graphs II", in Table 4 on page 260 the values a(30) = 7898574056034638 and a(31) = 30873429530206738 are incorrect due to the incorrect Gz(30) = 5876236938019300 and Gz(31) = 22974847474172100. - Wang Kai, Jun 05 2016

			For n = 3, there are 4 different graphic sequences possible: 0 0 0; 1 1 0; 2 1 1; 2 2 2. - Daan van Berkel (daan.v.berkel.1980(AT)gmail.com), Jun 25 2010
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(4) = 11 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)  (0,0,0,0)
           (1,1)  (0,1,1)  (0,0,1,1)
                  (1,1,2)  (0,1,1,2)
                  (2,2,2)  (0,2,2,2)
                           (1,1,1,1)
                           (1,1,1,3)
                           (1,1,2,2)
                           (1,2,2,3)
                           (2,2,2,2)
                           (2,2,3,3)
                           (3,3,3,3)
For example, the graph {{2,3},{2,4}} has degrees (0,2,1,1), so (0,1,1,2) is counted under a(4).
(End)

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
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).

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (Terms 1 through 31 were computed by various authors; terms 32 through 34 by Axel Kohnert and Wang Kai; terms 35 to 79 by Wang Kai)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995), #R11.
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
D. Dimitrov, Efficient computation of trees with minimal atom-bond connectivity index, arXiv:1305.1155 [cs.DM], 2013.
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapientiae, Inform. 3(2) (2011), 230-268.
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012), 260-288. - From _N. J. A. Sloane_, Feb 15 2013
A. Ivanyi and J. E. Schoenfield, Deciding football sequences, Acta Univ. Sapientiae, Informatica, 4, 1 (2012), 130-183. - From _N. J. A. Sloane_, Dec 22 2012 [Disclaimer: I am not one of the authors of this paper; I was unpleasantly surprised to find my name on it, as explained here. - _Jon E. Schoenfield_, Nov 26 2016]
Wang Kai, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms.
P. W. Mills, R. P. Rundle, V. M. Dwyer, T. Tilma, and S. J. Devitt, A proposal for an efficient quantum algorithm solving the graph isomorphism problem, arXiv 1711.09842, 2017.
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, Degree Sequence.
Eric Weisstein's World of Mathematics, Graphic Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
Index entries for sequences related to graphical partitions

Counting the positive partitions by sum gives A000569, ranked by A320922.
The version with half-loops is A029889, with covering case A339843.
The covering case (no zeros) is A095268.
Covering simple graphs are ranked by A309356 and A320458.
Non-graphical partitions are counted by A339617 and ranked by A339618.
The version with loops is A339844, with covering case A339845.
A006125 counts simple graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320921 counts connected graphical partitions.
A322353 counts factorizations into distinct semiprimes.
A339659 counts graphical partitions of 2n into k parts.
A339661 counts factorizations into distinct squarefree semiprimes.
Cf. A004250, A007717, A181819, A320894, A320923, A339560, A339561, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{2}]]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

G.f. = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 102*x^6 + 342*x^7 + 1213*x^8 + ...

a(n) ~ c * 4^n / n^(3/4) for some constant c > 0. Computational estimates suggest c ≈ 0.099094. - Tom Johnston, Jan 18 2023

More terms from Torsten Sillke, torsten.sillke(AT)lhsystems.com, using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
a(19) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 19 2007
a(20)-a(23) from Nathann Cohen, Jul 09 2011
a(24)-a(29) from Antal Iványi, Nov 15 2011
a(30) and a(31) corrected by Wang Kai, Jun 05 2016

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

A035363 Number of partitions of n into even parts.

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A320911 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes.

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A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

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A320912 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into distinct semiprimes.

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A320891 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes.

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A338900 Difference between the two prime indices of the n-th squarefree semiprime.

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A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

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

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