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
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[#]!={}]&]
A035363
Number of partitions of n into even parts.
Original entry on oeis.org
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
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.
Bisection (even part) gives the partition numbers
A000041.
Note: A-numbers of ranking sequences are in parentheses below.
The version for parts divisible by 3 instead of 2 is
A035377.
The Heinz numbers of these partitions are given by
A066207.
The ordered version (compositions) is
A077957 prepended by (1,0).
The multiplicative version (factorizations) is
A340785.
The following count partitions of even length:
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
A000569
Number of graphical partitions of 2n.
Original entry on oeis.org
1, 2, 5, 9, 17, 31, 54, 90, 151, 244, 387, 607, 933, 1420, 2136, 3173, 4657, 6799, 9803, 14048, 19956, 28179, 39467, 54996, 76104, 104802, 143481, 195485, 264941, 357635, 480408, 642723, 856398, 1136715, 1503172, 1980785
Offset: 1
a(2)=2: the graphical partitions of 4 are 2+1+1 and 1+1+1+1, corresponding to the degree sequences of the graphs V and ||.
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(1) = 1 through a(5) = 17 graphical partitions:
(11) (211) (222) (2222) (3322)
(1111) (2211) (3221) (22222)
(3111) (22211) (32221)
(21111) (32111) (33211)
(111111) (41111) (42211)
(221111) (222211)
(311111) (322111)
(2111111) (331111)
(11111111) (421111)
(511111)
(2221111)
(3211111)
(4111111)
(22111111)
(31111111)
(211111111)
(1111111111)
(End)
- T. D. Noe, Table of n, a(n) for n = 1..860. [Terms 1 through 110 were computed by Tiffany M. Barnes and Carla D. Savage; terms 111 through 585 were computed by Axel Kohnert; terms 586 to 860 by Wang Kai, Jun 05 2016; a typo of a(547) in Number of Graphical Partitions is corrected by Wang Kai, Aug 03 2016]
- Tiffany M. Barnes and Carla D. Savage, Efficient generation of graphical partitions Discrete Appl. Math. 78 (1997), no. 1-3, 17-26.
- T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
- K. Blum, Bounds on the Number of Graphical Partitions, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
- P. Erdős and T. Gallai, Graphs with a given degree of vertices, Mat. Lapok, 11 (1960), 264-274.
- P. Erdős and L. B. Richmond, On graphical partitions Combinatorica 13 (1993), no. 1, 57-63.
- Axel Kohnert, Dominance Order and Graphical Partitions, Electronic J. Combinatorics, 11 (2004).
- Gerard Sierksma and Han Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991), no. 2, 223-231.
- Eric Weisstein's World of Mathematics, Graphical partition.
- Gus Wiseman, Simple graphs realizing each of the a(6) = 31 graphical partitions of 12.
- Index entries for sequences related to graphical partitions
Cf.
A000070,
A000219,
A004250,
A004251,
A007717,
A025065,
A029889,
A095268,
A096373,
A147878,
A209816,
A320911,
A320921,
A320922.
-
<< MathWorld`Graphs`
Table[Count[RealizeDegreeSequence /@ Partitions[n], _Graph], {n, 2, 20, 2}]
(* second program *)
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,6}] (* Gus Wiseman, Oct 26 2018 *)
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
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)
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 *)
A320924
Heinz numbers of multigraphical partitions.
Original entry on oeis.org
1, 4, 9, 12, 16, 25, 27, 30, 36, 40, 48, 49, 63, 64, 70, 75, 81, 84, 90, 100, 108, 112, 120, 121, 144, 147, 154, 160, 165, 169, 175, 189, 192, 196, 198, 210, 220, 225, 243, 250, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 343, 351, 352, 360
Offset: 1
The sequence of all multigraphical partitions begins: (), (11), (22), (211), (1111), (33), (222), (321), (2211), (3111), (21111), (44), (422), (111111), (431), (332), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (221111).
From _Gus Wiseman_, May 23 2021: (Start)
The sequence of terms together with their prime indices and a multigraph realizing each begins:
1: () | {}
4: (11) | {{1,2}}
9: (22) | {{1,2},{1,2}}
12: (112) | {{1,3},{2,3}}
16: (1111) | {{1,2},{3,4}}
25: (33) | {{1,2},{1,2},{1,2}}
27: (222) | {{1,2},{1,3},{2,3}}
30: (123) | {{1,3},{2,3},{2,3}}
36: (1122) | {{1,2},{3,4},{3,4}}
40: (1113) | {{1,4},{2,4},{3,4}}
48: (11112) | {{1,2},{3,5},{4,5}}
49: (44) | {{1,2},{1,2},{1,2},{1,2}}
63: (224) | {{1,3},{1,3},{2,3},{2,3}}
(End)
These partitions are counted by
A209816.
The case with odd weights is
A322109.
The conjugate case of equality is
A340387.
The conjugate version with odd weights allowed is
A344291.
The conjugate opposite version is
A344292.
The opposite version with odd weights allowed is
A344296.
The conjugate opposite version with odd weights allowed is
A344414.
A000070 counts non-multigraphical partitions.
A025065 counts palindromic partitions.
A035363 counts partitions into even parts.
A110618 counts partitions that are the vertex-degrees of some set multipartition with no singletons.
A334201 adds up all prime indices except the greatest.
Cf.
A000041,
A000569,
A007717,
A096373,
A265640,
A283877,
A306005,
A318361,
A320459,
A320911,
A320922,
A320923,
A320925.
-
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],prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]!={}&]
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
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@@#&]!={}&]
A340387
Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.
Original entry on oeis.org
1, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 208, 243, 252, 264, 270, 280, 300, 544, 624, 729, 756, 784, 792, 810, 840, 880, 900, 1000, 1216, 1632, 1872, 2080, 2187, 2268, 2352, 2376, 2430, 2464, 2520, 2640, 2700, 2800, 2944, 3000, 3648, 4896, 5440, 5616
Offset: 1
The sequence of terms together with their prime indices begins:
1: {}
3: {2}
9: {2,2}
10: {1,3}
27: {2,2,2}
28: {1,1,4}
30: {1,2,3}
81: {2,2,2,2}
84: {1,1,2,4}
88: {1,1,1,5}
90: {1,2,2,3}
100: {1,1,3,3}
208: {1,1,1,1,6}
243: {2,2,2,2,2}
252: {1,1,2,2,4}
Partitions of 2n into n parts are counted by
A000041.
The number of prime indices alone is
A001222.
The sum of prime indices alone is
A056239.
Allowing sum to be any multiple of length gives
A067538, ranked by
A316413.
A301987 lists numbers whose sum of prime indices equals their product, with nonprime case
A301988.
Cf.
A000720,
A001221,
A001414,
A006125,
A006129,
A112798,
A316428,
A320911,
A325037,
A325044,
A330950,
A331385,
A331416.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]
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
9000 is in the sequence and can be factored in either of two ways: (4*6*15*25) or (4*9*10*25).
-
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[#]!={}]&]
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
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.
A004526 counts 2-part partitions, with strict case
A140106 (shifted left).
A065516 gives first differences of semiprimes.
A166237 gives first differences of squarefree semiprimes.
Cf.
A000040,
A056239,
A112798,
A167171,
A320656,
A320891,
A320894,
A320911,
A338898,
A338905,
A338908.
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
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).
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.
A002100 counts partitions into squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by
A339657.
A339659 counts graphical partitions of 2n into k parts.
The following count partitions of even length and give their Heinz numbers:
-
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}]
Showing 1-10 of 48 results.
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