A338902 -id:A338902 - cp's OEIS frontend

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.

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

A112141 Product of the first n semiprimes.

Original entry on oeis.org

4, 24, 216, 2160, 30240, 453600, 9525600, 209563200, 5239080000, 136216080000, 4495130640000, 152834441760000, 5349205461600000, 203269807540800000, 7927522494091200000, 364666034728195200000, 17868635701681564800000, 911300420785759804800000

Offset: 1

Jonathan Vos Post, Nov 28 2005

Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.

			a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
From _Gus Wiseman_, Dec 06 2020: (Start)
The sequence of terms together with their prime signatures begins:
                        4: (2)
                       24: (3,1)
                      216: (3,3)
                     2160: (4,3,1)
                    30240: (5,3,1,1)
                   453600: (5,4,2,1)
                  9525600: (5,5,2,2)
                209563200: (6,5,2,2,1)
               5239080000: (6,5,4,2,1)
             136216080000: (7,5,4,2,1,1)
            4495130640000: (7,6,4,2,2,1)
          152834441760000: (8,6,4,2,2,1,1)
         5349205461600000: (8,6,5,3,2,1,1)
       203269807540800000: (9,6,5,3,2,1,1,1)
      7927522494091200000: (9,7,5,3,2,2,1,1)
    364666034728195200000: (10,7,5,3,2,2,1,1,1)
  17868635701681564800000: (10,7,5,5,2,2,1,1,1)
(End)

T. D. Noe, Table of n, a(n) for n = 1..100

Partial sums of semiprimes are A062198.
First differences of semiprimes are A065516.
A000040 lists primes, with partial products A002110 (primorials).
A000142 lists factorials, with partial products A000178 (superfactorials).
A001358 lists semiprimes, with partial products A112141 (this sequence).
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial products A339191.
A101048 counts partitions into semiprimes (restricted: A338902).
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
Cf. A001222, A084126, A084127, A115392, A168472, A320732, A320892.

A112141 := proc(n)
    mul(A001358(i),i=1..n) ;
end proc:
seq(A112141(n),n=1..10) ; # R. J. Mathar, Jun 30 2020

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Times @@ NestList[ NextSemiPrime@# &, 2^2, n - 1]; Array[f, 18] (* Robert G. Wilson v, Jun 13 2013 *)
FoldList[Times,Select[Range[30],PrimeOmega[#]==2&]] (* Gus Wiseman, Dec 06 2020 *)

a(n)=my(v=vector(n),i,k=3);while(iCharles R Greathouse IV, Apr 04 2013

from sympy import factorint
def aupton(terms):
    alst, k, p = [], 1, 1
    while len(alst) < terms:
        if sum(factorint(k).values()) == 2:
            p *= k
            alst.append(p)
        k += 1
    return alst
print(aupton(18)) # Michael S. Branicky, Aug 31 2021

a(n) = Product_{i=1..n} A001358(i).

A001222(a(n)) = 2*n.

A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 6, 8, 12, 16, 21, 28, 37, 49, 64, 80, 104, 135, 169, 216, 268, 341, 420, 527, 654, 809, 991, 1218, 1488, 1828, 2213, 2687, 3262, 3934, 4754, 5702, 6849, 8200, 9819, 11693, 13937, 16562, 19659, 23262, 27577, 32493, 38341, 45112, 53059, 62265

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).

			The a(2) = 1 through a(10) = 16 partitions:
  (11)  (21)  (22)  (32)    (33)    (43)    (44)    (54)      (55)
              (31)  (41)    (42)    (52)    (53)    (63)      (64)
                    (2111)  (51)    (61)    (62)    (72)      (73)
                            (2211)  (2221)  (71)    (81)      (82)
                            (3111)  (3211)  (3221)  (3222)    (91)
                                    (4111)  (3311)  (3321)    (3322)
                                            (4211)  (4221)    (3331)
                                            (5111)  (4311)    (4222)
                                                    (5211)    (4321)
                                                    (6111)    (4411)
                                                    (222111)  (5221)
                                                    (321111)  (5311)
                                                              (6211)
                                                              (7111)
                                                              (322111)
                                                              (421111)
For example, the partition (4,2,1,1,1,1) can be partitioned into {{1,1},{1,2},{1,4}}, and thus is counted under a(10).

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 gives the Heinz numbers of these partitions.
A338915 counts the complement in even-length partitions.
A339563 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.
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).
- A338915 cannot be partitioned into distinct pairs (A320892).
- 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, A320921, A338898, A338902, A339564.

stfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],stfs[Times@@Prime/@#]!={}&]],{n,0,20}]

A027187(n) = a(n) + A338915(n).

More terms from Jinyuan Wang, Feb 14 2025

A339655 Number of non-loop-graphical integer partitions of 2n.

Original entry on oeis.org

0, 0, 1, 3, 7, 14, 28, 51, 91, 156, 260, 425, 680, 1068, 1654, 2524, 3802, 5668, 8350, 12190, 17634, 25306, 36011, 50902, 71441, 99642

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(2) = 1 through a(5) = 14 partitions (A = 10):
  (4)  (6)    (8)      (A)
       (4,2)  (4,4)    (5,5)
       (5,1)  (5,3)    (6,4)
              (6,2)    (7,3)
              (7,1)    (8,2)
              (5,2,1)  (9,1)
              (6,1,1)  (5,3,2)
                       (5,4,1)
                       (6,2,2)
                       (6,3,1)
                       (7,2,1)
                       (8,1,1)
                       (6,2,1,1)
                       (7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
  {{1,1},{1,1},{1,2},{2,2},{3,3}}
  {{1,1},{1,1},{1,2},{2,3},{2,3}}
  {{1,1},{1,1},{1,3},{2,2},{2,3}}
  {{1,1},{1,2},{1,2},{1,2},{3,3}}
  {{1,1},{1,2},{1,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{1,3},{2,2}}
  {{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).

Eric Weisstein's World of Mathematics, Graphical partition.

A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A320461 ranks normal loop-graphs.
A322661 counts covering loop-graphs.
A320655 counts factorizations into 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).
- A339655 (this sequence) 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).
Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]

A058696(n) = a(n) + A339656(n).

a(7)-a(25) from Andrew Howroyd, Jan 10 2024

A339656 Number of loop-graphical integer partitions of 2n.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 49, 84, 140, 229, 367, 577, 895, 1368, 2064, 3080, 4547, 6642, 9627, 13825, 19704, 27868, 39164, 54656, 75832, 104584

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the unordered prime signature of n is loop-graphical.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)    (2,2)      (3,3)          (3,3,2)
      (1,1)  (3,1)      (2,2,2)        (4,2,2)
             (2,1,1)    (3,2,1)        (4,3,1)
             (1,1,1,1)  (4,1,1)        (2,2,2,2)
                        (2,2,1,1)      (3,2,2,1)
                        (3,1,1,1)      (3,3,1,1)
                        (2,1,1,1,1)    (4,2,1,1)
                        (1,1,1,1,1,1)  (5,1,1,1)
                                       (2,2,2,1,1)
                                       (3,2,1,1,1)
                                       (4,1,1,1,1)
                                       (2,2,1,1,1,1)
                                       (3,1,1,1,1,1)
                                       (2,1,1,1,1,1,1)
                                       (1,1,1,1,1,1,1,1)
For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
  {{1,1},{2,2},{3,4}}
  {{1,1},{2,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}
  {{1,3},{1,4},{2,2}},
so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
  {{1,1},{1,1},{2,2}}
  {{1,1},{1,2},{1,2}},
but neither of these is a loop-graph, so z is not counted under a(3).

Eric Weisstein's World of Mathematics, Graphical partition.

A339658 ranks these partitions.
A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A062740 counts labeled connected loop-graphs.
A320461 ranks normal loop-graphs.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A322661 counts covering loop-graphs.
A339845 counts the same partitions by length, or A339844 with zeros.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 [this sequence] counts loop-graphical partitions (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).
- 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, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]!={}&]],{n,0,5}]

A058696(n) = a(n) + A339655(n).

a(8)-a(25) from Andrew Howroyd, Jan 10 2024

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

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

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

A339191 Partial products of squarefree semiprimes (A006881).

Original entry on oeis.org

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000

Offset: 1

Gus Wiseman, Nov 30 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers.

Do all terms belong to A242031 (weakly decreasing prime signature)?

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

cp's OEIS Frontend

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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A112141 Product of the first n semiprimes.

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A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

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A339655 Number of non-loop-graphical integer partitions of 2n.

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A339656 Number of loop-graphical integer partitions of 2n.

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A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

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A339191 Partial products of squarefree semiprimes (A006881).

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