A306844
Number of anti-transitive rooted trees with n nodes.
Original entry on oeis.org
1, 1, 2, 3, 7, 14, 36, 83, 212, 532, 1379, 3577, 9444, 25019, 66943, 179994, 487031, 1323706, 3614622, 9907911
Offset: 1
The a(1) = 1 through a(6) = 14 anti-transitive rooted trees:
o (o) (oo) (ooo) (oooo) (ooooo)
((o)) ((oo)) ((ooo)) ((oooo))
(((o))) (((oo))) (((ooo)))
((o)(o)) ((o)(oo))
((o(o))) ((o(oo)))
(o((o))) ((oo(o)))
((((o)))) (o((oo)))
(oo((o)))
((((oo))))
(((o)(o)))
(((o(o))))
((o((o))))
(o(((o))))
(((((o)))))
Cf.
A324694,
A324695,
A324738,
A324741,
A324743,
A324751,
A324754,
A324756,
A324758,
A324759,
A324764.
-
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@#,#]=={}&]],{n,10}]
A324844
Number of unlabeled rooted trees with n nodes where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
Original entry on oeis.org
1, 1, 2, 3, 7, 13, 32, 71, 170, 406, 1002, 2469, 6204, 15644, 39871, 102116, 263325, 682079, 1775600, 4640220
Offset: 1
The a(1) = 1 through a(6) = 13 rooted trees:
o (o) (oo) (ooo) (oooo) (ooooo)
((o)) ((oo)) ((ooo)) ((oooo))
(((o))) (o(oo)) (o(ooo))
(((oo))) (((ooo)))
((o)(o)) ((o)(oo))
(o((o))) ((o(oo)))
((((o)))) (o((oo)))
(oo((o)))
((((oo))))
(((o)(o)))
((o((o))))
(o(((o))))
(((((o)))))
The Matula-Goebel numbers of these trees are given by
A324845.
Cf.
A324694,
A324738,
A324744,
A324749,
A324754,
A324759,
A324765,
A324768,
A324838,
A324843,
A324846,
A324847,
A324848,
A324849.
-
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rallt[n_]:=Select[Union[Sort/@Join@@(Tuples[rallt/@#]&/@IntegerPartitions[n-1])],And@@Table[!submultQ[b,#],{b,DeleteCases[#,{}]}]&];
Table[Length[rallt[n]],{n,10}]
A324744
Number of maximal subsets of {1...n} containing no element whose prime indices all belong to the subset.
Original entry on oeis.org
1, 1, 2, 2, 3, 4, 4, 5, 6, 8, 8, 11, 11, 22, 22, 22, 22, 28, 28, 44, 44, 52, 52, 76, 76, 88, 88, 96, 96, 184, 184, 240, 240, 264, 264, 296, 296, 592, 592, 592, 592, 728, 728, 1456, 1456, 1456, 1456, 2912, 2912, 3168, 3168, 3168, 3168, 5568, 5568, 5568, 5568
Offset: 0
The a(1) = 1 through a(8) = 6 maximal subsets:
{1} {1} {2} {1,3} {1,3} {1,3,6} {3,4,6} {1,3,6,7}
{2} {1,3} {2,4} {1,5} {1,5,6} {1,3,6,7} {1,5,6,7}
{3,4} {3,4} {3,4,6} {1,5,6,7} {3,4,6,8}
{2,4,5} {2,4,5,6} {2,4,5,6} {3,6,7,8}
{2,5,6,7} {2,4,5,6,8}
{2,5,6,7,8}
The non-maximal case is
A324738. The case for subsets of {2...n} is
A324762.
Cf.
A000720,
A001462,
A007097,
A076078,
A084422,
A085945,
A112798,
A276625,
A290822,
A304360,
A306844,
A320426,
A324764.
-
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,0,10}]
-
pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019
A324738
Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.
Original entry on oeis.org
1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784
Offset: 0
The a(0) = 1 through a(6) = 26 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{1,3} {4} {4} {4}
{1,3} {5} {5}
{2,4} {1,3} {6}
{3,4} {1,5} {1,3}
{2,4} {1,5}
{2,5} {1,6}
{3,4} {2,4}
{4,5} {2,5}
{2,4,5} {2,6}
{3,4}
{3,6}
{4,5}
{4,6}
{5,6}
{1,3,6}
{1,5,6}
{2,4,5}
{2,4,6}
{2,5,6}
{3,4,6}
{4,5,6}
{2,4,5,6}
The maximal case is
A324744. The case of subsets of {2...n} is
A324739. The strict integer partition version is
A324749. The integer partition version is
A324754. The Heinz number version is
A324759. An infinite version is
A324694.
Cf.
A000720,
A001221,
A001462,
A007097,
A076078,
A084422,
A085945,
A112798,
A276625,
A279861,
A290689,
A290822,
A304360,
A306844.
-
Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]
-
pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019
A324759
Heinz numbers of integer partitions containing no part > 1 whose prime indices all belong to the partition.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 74, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93
Offset: 1
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
10: {1,3}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
20: {1,1,3}
21: {2,4}
22: {1,5}
23: {9}
25: {3,3}
26: {1,6}
The subset version is
A324738, with maximal case
A324744. The strict integer partition version is
A324749. The integer partition version is
A324754. An infinite version is
A324694.
Cf.
A000720,
A001221,
A007097,
A056239,
A112798,
A276625,
A289509,
A290822,
A306844,
A324695,
A324750,
A324755,
A324760.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[DeleteCases[primeMS[#],1],k_/;SubsetQ[primeMS[#],primeMS[k]]]&]
A324762
Number of maximal subsets of {2...n} containing no element whose prime indices all belong to the subset.
Original entry on oeis.org
1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 40, 40, 52, 52, 64, 64, 72, 72, 144, 144, 176, 176, 200, 200, 232, 232, 464, 464, 464, 464, 536, 536, 1072, 1072, 1072, 1072, 2144, 2144, 2400, 2400, 2400, 2400, 4800, 4800, 4800, 4800, 4800
Offset: 1
The a(2) = 1 through a(9) = 6 maximal subsets:
{2} {2} {2,4} {3,4} {3,4,6} {3,4,6} {3,4,6,8} {2,4,5,6,8}
{3} {3,4} {2,4,5} {2,4,5,6} {3,6,7} {3,6,7,8} {2,5,6,7,8}
{2,4,5,6} {2,4,5,6,8} {3,4,6,8,9}
{2,5,6,7} {2,5,6,7,8} {3,6,7,8,9}
{4,5,6,8,9}
{5,6,7,8,9}
The non-maximal version is
A324739.
The version for subsets of {1...n} is
A324744.
-
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,10}]
-
pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019
A324749
Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462
Offset: 0
The a(0) = 1 through a(10) = 6 strict integer partitions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,1) (4,2) (4,3) (6,2) (5,4) (6,4)
(5,1) (5,2) (7,1) (6,3) (7,3)
(6,1) (7,2) (8,2)
(9,1)
(6,3,1)
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]
A324845
Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.
Original entry on oeis.org
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106
Offset: 1
The sequence of terms together with their Matula-Goebel numbers begins:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
7: ((oo))
8: (ooo)
9: ((o)(o))
10: (o((o)))
11: ((((o))))
14: (o(oo))
16: (oooo)
17: (((oo)))
19: ((ooo))
20: (oo((o)))
21: ((o)(oo))
22: (o(((o))))
23: (((o)(o)))
25: (((o))((o)))
27: ((o)(o)(o))
Cf.
A324694,
A324738,
A324744,
A324749,
A324754,
A324759,
A324765,
A324768,
A324838,
A324842,
A324844,
A324846,
A324847,
A324849.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]
Showing 1-8 of 8 results.
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