A320913
Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into squarefree semiprimes (A320891) but can be factored into distinct semiprimes (A320912).
Original entry on oeis.org
4, 9, 24, 25, 40, 49, 54, 56, 88, 104, 121, 135, 136, 152, 169, 184, 189, 232, 240, 248, 250, 289, 296, 297, 328, 336, 344, 351, 361, 375, 376, 424, 459, 472, 488, 513, 528, 529, 536, 560, 568, 584, 621, 624, 632, 664, 686, 712, 776, 783, 808, 810, 816, 824
Offset: 1
Cf.
A001055,
A001222,
A001358,
A005117,
A006881,
A007717,
A025487,
A028260,
A320655,
A320656,
A320891,
A320892,
A320893,
A320894,
A320911,
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]&]}]];
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[#]!={},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
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
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).
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[#]!={}]&]
A338898
Concatenated sequence of prime indices of semiprimes (A001358).
Original entry on oeis.org
1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 3, 3, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 4, 4, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 5, 5, 1, 18, 2
Offset: 1
The sequence of semiprimes together with their prime indices begins:
4: {1,1} 46: {1,9} 91: {4,6} 141: {2,15}
6: {1,2} 49: {4,4} 93: {2,11} 142: {1,20}
9: {2,2} 51: {2,7} 94: {1,15} 143: {5,6}
10: {1,3} 55: {3,5} 95: {3,8} 145: {3,10}
14: {1,4} 57: {2,8} 106: {1,16} 146: {1,21}
15: {2,3} 58: {1,10} 111: {2,12} 155: {3,11}
21: {2,4} 62: {1,11} 115: {3,9} 158: {1,22}
22: {1,5} 65: {3,6} 118: {1,17} 159: {2,16}
25: {3,3} 69: {2,9} 119: {4,7} 161: {4,9}
26: {1,6} 74: {1,12} 121: {5,5} 166: {1,23}
33: {2,5} 77: {4,5} 122: {1,18} 169: {6,6}
34: {1,7} 82: {1,13} 123: {2,13} 177: {2,17}
35: {3,4} 85: {3,7} 129: {2,14} 178: {1,24}
38: {1,8} 86: {1,14} 133: {4,8} 183: {2,18}
39: {2,6} 87: {2,10} 134: {1,19} 185: {3,12}
A112798 restricted to rows of length 2 gives this triangle.
A115392 is the row number for the first appearance of each positive integer.
A001221 counts a number's distinct prime indices.
A001222 counts a number's prime indices.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes.
A065516 gives first differences of semiprimes.
A320655 counts factorizations into semiprimes.
Cf.
A056239,
A101048,
A320892,
A320912,
A338900,
A338901,
A338904,
A338906,
A338907,
A338910,
A338911.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]
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}]
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
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.
A309356 is a kind of universal embedding.
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.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
The following count partitions of even length and give their Heinz numbers:
-
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[#]!={}&]
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[#]!={}]&]
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
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)
A096373 counts the complement in even-length partitions.
A320911 gives the Heinz numbers of these partitions.
A339562 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by
A339620.
A002100 counts partitions into squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by
A339657.
The following count partitions of even length and give their Heinz numbers:
-
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]
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
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}}
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:
-
A339617 [this sequence] counts non-graphical partitions of 2n (
A339618).
The following count partitions of even length and give their Heinz numbers:
-
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}]
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
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).
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.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by
A339657.
The following count partitions of even length and give their Heinz numbers:
-
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}]
Showing 1-10 of 30 results.
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