A320656 -id:A320656 - cp's OEIS frontend

A174725 a(n) = (A074206(n) + A008683(n))/2.

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2

Offset: 1

Mats Granvik, Mar 28 2010

nonn

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
  (2*6)  (3*8)      (5*6)   (4*8)      (4*9)
  (3*4)  (4*6)      (6*5)   (8*4)      (6*6)
  (4*3)  (6*4)      (10*3)  (16*2)     (9*4)
  (6*2)  (8*3)      (15*2)  (2*16)     (12*3)
         (12*2)     (2*15)  (2*2*2*4)  (18*2)
         (2*12)     (3*10)  (2*2*4*2)  (2*18)
         (2*2*2*3)          (2*4*2*2)  (3*12)
         (2*2*3*2)          (4*2*2*2)  (2*2*3*3)
         (2*3*2*2)                     (2*3*2*3)
         (3*2*2*2)                     (2*3*3*2)
                                       (3*2*2*3)
                                       (3*2*3*2)
                                       (3*3*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

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.

A320893 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes (A320911) but cannot be factored into distinct semiprimes (A320892).

Original entry on oeis.org

1296, 7776, 10000, 12960, 18144, 19440, 21600, 27216, 28512, 33696, 36000, 38416, 42336, 42768, 44064, 46656, 48600, 49248, 50544, 50625, 59616, 60000, 66096, 73872, 75168, 77760, 80352, 89424, 95256, 95904, 98784, 100000

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

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

Cf. A001055, A001358, A005117, A006881, A007717, A028260, A318871, A318953, A320655, A320656, A320891, A320892, A320894, A320911, A320912, A320913.

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]&]}]];
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[10000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={},sqfsemfacs[#]!={}]&]

A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 4, 6, 9, 11, 16, 23, 29, 39, 53, 69, 90, 118, 150, 195, 249, 315, 398, 506, 629, 789, 982, 1219, 1504, 1860, 2277, 2798, 3413, 4161, 5051, 6137, 7406, 8948, 10765, 12943, 15503, 18571, 22153, 26432, 31432, 37352, 44268, 52444, 61944, 73141

Offset: 0

Gus Wiseman, Dec 09 2020

nonn

These are also integer partitions that can be partitioned into not necessarily distinct edges (pairs of distinct parts). For example, (3,3,2,2) can be partitioned as {{2,3},{2,3}}, so is counted under a(10), but (4,2,2,2) and (4,2,1,1,1,1) cannot be partitioned into edges. The multiplicities of such a partition form a multigraphical partition (A209816, A320924).

			The a(3) = 1 through a(10) = 11 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
              (41)  (51)    (52)    (62)    (63)      (73)
                    (2211)  (61)    (71)    (72)      (82)
                            (3211)  (3221)  (81)      (91)
                                    (3311)  (3321)    (3322)
                                    (4211)  (4221)    (4321)
                                            (4311)    (4411)
                                            (5211)    (5221)
                                            (222111)  (5311)
                                                      (6211)
                                                      (322111)

Eric Weisstein's World of Mathematics, Graphical partition.

A096373 counts the complement in even-length partitions.
A320911 gives the Heinz numbers of these partitions.
A339560 is the strict case.
A339562 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.
A320656 counts factorizations into squarefree semiprimes.
A320921 counts connected graphical partitions, ranked by A320923.
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.
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).
- 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. A001055, A001221, A005117, A007717, A030229, A320655, A322353, A338899, A338903.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]

A027187(n) = a(n) + A096373(n).

A339617 Number of non-graphical integer partitions of 2n.

Original entry on oeis.org

0, 1, 3, 6, 13, 25, 46, 81, 141, 234, 383, 615, 968, 1503, 2298, 3468, 5176, 7653, 11178, 16212, 23290, 33218, 46996, 66091, 92277, 128122, 176787, 242674, 331338, 450279, 608832, 819748, 1098907, 1467122, 1951020, 2584796, 3411998

Offset: 0

Gus Wiseman, Dec 13 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. See A209816 for multigraphical partitions, A000070 for non-multigraphical partitions. Graphical partitions are counted by A000569.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices 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 a(1) = 1 through a(4) = 13 partitions:
  (2)  (4)    (6)      (8)
       (2,2)  (3,3)    (4,4)
       (3,1)  (4,2)    (5,3)
              (5,1)    (6,2)
              (3,2,1)  (7,1)
              (4,1,1)  (3,3,2)
                       (4,2,2)
                       (4,3,1)
                       (5,2,1)
                       (6,1,1)
                       (3,3,1,1)
                       (4,2,1,1)
                       (5,1,1,1)
For example, the partition (2,2,2,2) is not counted under a(4) because there are three possible graphs with the prescribed degrees:
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Graphical partition.

A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
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).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 [this sequence] 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).
Cf. A001055, A007717, A025065, A320921, A320922, A338899, A339564, A339619, A339660, A339661.

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[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[prptns[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) + A000569(n) = A000041(2*n).

A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 20 2021

nonn

I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021

			The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
  3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
  2*2*2*2*2*6   2*2*3*4*6*6   2*3*4*4*6*6         2*2*2*2*2*6*6*8
                2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
                              2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
                              2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
                                                  2*2*2*2*2*2*2*8*9
                                                  2*2*2*2*2*2*4*4*9

Antti Karttunen, Table of n, a(n) for n = 1..100000

The co-balanced version is A340596.
Positions of nonzero terms are A340597.
The case of powers of two is A340611.
Taking maximum Omega instead of maximum factor gives A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==Max[#]&]],{n,100}]

A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024

A338915 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 4, 2, 6, 6, 12, 12, 20, 22, 38, 42, 60, 73, 101, 124, 164, 203, 266, 319, 415, 507, 649, 786, 983, 1198, 1499, 1797, 2234, 2673, 3303, 3952, 4826, 5753, 6999, 8330, 10051, 11943, 14357, 16956, 20322, 23997, 28568, 33657, 39897, 46879

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).

			The a(7) = 1 through a(12) = 12 partitions:
  211111  2222      411111    222211      222221      3333
          221111    21111111  331111      611111      222222
          311111              511111      22211111    441111
          11111111            22111111    32111111    711111
                              31111111    41111111    22221111
                              1111111111  2111111111  32211111
                                                      33111111
                                                      42111111
                                                      51111111
                                                      2211111111
                                                      3111111111
                                                      111111111111
For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}}
None of these is strict, so y is counted under a(12).

Eric Weisstein's World of Mathematics, Graphical partition.

The Heinz numbers of these partitions are A320892.
The complement in even-length partitions is A338916.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct 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.
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).
- 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. A001055, A007717, A025065, A320656, A320732, A320893, A338898, A338902.

smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]

A027187(n) = a(n) + A338916(n).

More terms from Jinyuan Wang, Feb 14 2025

A340654 Number of cross-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 5, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).

			The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
  2*6   4*6     4*9     2*4*9     4*4*9       8*30
  3*4   2*2*6   6*6     2*6*6     4*6*6       12*20
        2*3*4   2*2*9   3*4*6     2*2*4*9     5*6*8
                2*3*6   2*2*2*9   2*2*6*6     2*4*30
                3*3*4   2*2*3*6   2*3*4*6     2*6*20
                        2*3*3*4   3*3*4*4     2*8*15
                                  2*2*2*2*9   3*4*20
                                  2*2*2*3*6   3*8*10
                                  2*2*3*3*4   4*5*12
                                              2*10*12
                                              2*3*5*8
                                              2*2*2*30
                                              2*2*3*20
                                              2*2*5*12

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340651.
The balanced version is A340653.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A324518, A339846, A339890, A340608.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],#=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,100}]

A340654(n, m=n, om=omega(n),mbo=0) = if(1==n,(mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo,bigomega(d)))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(105) by Antti Karttunen, Jun 19 2024

A339618 Heinz numbers of non-graphical integer partitions of even numbers.

Original entry on oeis.org

3, 7, 9, 10, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 147

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the unordered prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         43: {14}        79: {22}
      7: {4}         46: {1,9}       82: {1,13}
      9: {2,2}       49: {4,4}       84: {1,1,2,4}
     10: {1,3}       52: {1,1,6}     85: {3,7}
     13: {6}         53: {16}        87: {2,10}
     19: {8}         55: {3,5}       88: {1,1,1,5}
     21: {2,4}       57: {2,8}       89: {24}
     22: {1,5}       61: {18}        91: {4,6}
     25: {3,3}       62: {1,11}      94: {1,15}
     28: {1,1,4}     63: {2,2,4}    100: {1,1,3,3}
     29: {10}        66: {1,2,5}    101: {26}
     30: {1,2,3}     70: {1,3,4}    102: {1,2,7}
     34: {1,7}       71: {20}       107: {28}
     37: {12}        75: {2,3,3}    111: {2,12}
     39: {2,6}       76: {1,1,8}    113: {30}
For example, there are three possible multigraphs with degrees (1,1,3,3):
  {{1,2},{1,2},{1,2},{3,4}}
  {{1,2},{1,2},{1,3},{2,4}}
  {{1,2},{1,2},{1,4},{2,3}}.
Since none of these is a graph, the Heinz number 100 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A181819 applied to A320894 gives this sequence.
A300061 is a superset.
A339617 counts these partitions.
A320922 ranks the complement, counted by A000569.
A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
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).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618 [this sequence]).
- 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).
Cf. A001358, A007717, A050326, A320923, A338899.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&strs[Times@@Prime/@nrmptn[#]]=={}&]

Equals A300061 \ A320922.

For all n, A181821(a(n)) and A304660(a(n)) belong to A320894.

cp's OEIS Frontend

A174725 a(n) = (A074206(n) + A008683(n))/2.

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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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A339561 Products of distinct squarefree semiprimes.

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A320893 Numbers with an even number of prime factors (counted with multiplicity) that can be factored into squarefree semiprimes (A320911) but cannot be factored into distinct semiprimes (A320892).

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A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

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A339617 Number of non-graphical integer partitions of 2n.

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A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

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A338915 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.

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