A324743
Number of maximal subsets of {1...n} containing no prime indices of the elements.
Original entry on oeis.org
1, 1, 2, 2, 3, 4, 5, 8, 8, 8, 8, 12, 12, 18, 18, 19, 19, 30, 30, 54, 54, 54, 54, 96, 96, 96, 96, 96, 96, 156, 156, 244, 244, 248, 248, 248, 248, 440, 440, 440, 440, 688, 688, 1120, 1120, 1120, 1120, 1864, 1864, 1864, 1864, 1864, 1864, 3664, 3664, 3664, 3664, 3664
Offset: 0
The a(0) = 1 through a(8) = 8 maximal subsets:
{} {1} {1} {2} {1,3} {1,3} {1,3} {1,3,7} {1,3,7}
{2} {1,3} {2,4} {1,5} {1,5} {1,5,7} {1,5,7}
{3,4} {3,4} {2,4,5} {2,4,5} {2,4,5,8}
{2,4,5} {3,4,6} {2,5,7} {2,5,7,8}
{4,5,6} {3,4,6} {3,4,6,8}
{3,6,7} {3,6,7,8}
{4,5,6} {4,5,6,8}
{5,6,7} {5,6,7,8}
An example for n = 15 is {1,5,7,9,13,15}, with prime indices:
1: {}
5: {3}
7: {4}
9: {2,2}
13: {6}
15: {2,3}
None of these prime indices {2,3,4,6} belong to the subset, as required.
The non-maximal case is
A324741. The case for subsets of {2...n} is
A324763.
Cf.
A000720,
A001462,
A007097,
A084422,
A085945,
A112798,
A276625,
A290689,
A290822,
A304360,
A306844,
A320426,
A324764.
-
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{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, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019
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}]
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
A324838
Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.
Original entry on oeis.org
1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288
Offset: 1
The a(1) = 1 through a(6) = 10 rooted trees:
o ((o)) ((oo)) ((ooo)) ((oooo))
(((o))) (((oo))) (((ooo)))
((o)(o)) ((o)(oo))
((o(o))) ((o(oo)))
((((o)))) ((oo(o)))
((((oo))))
(((o)(o)))
(((o(o))))
((o((o))))
(((((o)))))
Cf.
A324694,
A324696,
A324704,
A324738,
A324744,
A324758,
A324759,
A324765,
A324768,
A324771,
A324839,
A324840,
A324844,
A324846.
-
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]
A324755
Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
Original entry on oeis.org
1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 10, 7, 16, 14, 23, 23, 35, 34, 53, 54, 75, 80, 112, 115, 160, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011
Offset: 0
The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
(2) (3) (4) (5) (6) (7) (8) (9) (A)
(22) (33) (43) (44) (54) (55)
(42) (52) (62) (63) (64)
(222) (422) (72) (73)
(2222) (333) (82)
(522) (433)
(442)
(622)
(4222)
(22222)
Cf.
A000837,
A001462,
A051424,
A112798,
A276625,
A290822,
A304360,
A306844,
A324695,
A324696,
A324744.
-
Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]
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]]]&]
A324750
Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221
Offset: 0
The a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
2 3 4 5 6 7 8 9 A B C D E F G H
42 43 62 54 64 65 75 76 86 87 97 98
52 63 73 83 84 85 95 96 A6 A7
72 82 542 93 94 A4 A5 C4 B6
A2 A3 B3 B4 D3 C5
642 B2 C2 C3 E2 D4
643 752 D2 763 E3
652 842 654 862 F2
762 943 854
843 A42 863
852 872
A43
A52
B42
6542
Cf.
A000720,
A001462,
A007097,
A074971,
A078374,
A112798,
A276625,
A290822,
A304360,
A305713,
A306844.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]
A324754
Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.
Original entry on oeis.org
1, 1, 2, 2, 4, 3, 7, 8, 11, 12, 19, 19, 30, 34, 46, 50, 71, 76, 104, 119, 151, 171, 225, 247, 315, 360, 446, 504, 629, 703, 867, 986, 1192, 1346, 1636, 1837, 2204, 2500, 2965, 3348, 3980, 4475, 5276, 5963, 6973, 7852, 9194, 10335, 12009, 13536, 15650, 17589
Offset: 0
The a(1) = 1 through a(8) = 11 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (43) (44)
(31) (11111) (42) (52) (62)
(1111) (51) (61) (71)
(222) (331) (422)
(3111) (511) (611)
(111111) (31111) (2222)
(1111111) (3311)
(5111)
(311111)
(11111111)
Cf.
A000837,
A001462,
A007097,
A051424,
A112798,
A276625,
A290822,
A304360,
A306844,
A324695,
A324750,
A324755,
A324760.
-
Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]
A324760
Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.
Original entry on oeis.org
1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139
Offset: 1
The sequence of terms together with their prime indices begins:
1: {}
3: {2}
5: {3}
7: {4}
9: {2,2}
11: {5}
13: {6}
17: {7}
19: {8}
21: {2,4}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
33: {2,5}
35: {3,4}
37: {12}
39: {2,6}
41: {13}
The subset version is
A324739, with maximal case
A324762. The strict integer partition version is
A324750. The integer partition version is
A324755. An infinite version is
A324694.
Cf.
A000720,
A001221,
A007097,
A056239,
A112798,
A289509,
A290822,
A306844,
A324695,
A324696,
A324737,
A324744.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[primeMS[#],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
Showing 1-10 of 14 results.
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