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
A324763
Number of maximal subsets of {2...n} containing no prime indices of the elements.
Original entry on oeis.org
1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600
Offset: 1
The a(1) = 1 through a(9) = 6 subsets:
{} {2} {2} {2,4} {3,4} {2,4,5} {2,4,5} {2,4,5,8} {2,4,5,8}
{3} {3,4} {2,4,5} {3,4,6} {2,5,7} {2,5,7,8} {2,5,7,8}
{4,5,6} {3,4,6} {3,4,6,8} {3,4,6,8,9}
{3,6,7} {3,6,7,8} {3,6,7,8,9}
{4,5,6} {4,5,6,8} {4,5,6,8,9}
{5,6,7} {5,6,7,8} {5,6,7,8,9}
The non-maximal version is
A324742.
The version for subsets of {1...n} is
A324741.
-
maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{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-1, k, pset(k+1)>>1), 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
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}]
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]]]&]
A324739
Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.
Original entry on oeis.org
1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376
Offset: 1
The a(1) = 1 through a(6) = 20 subsets:
{} {} {} {} {} {}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{4} {4} {4}
{2,4} {5} {5}
{3,4} {2,4} {6}
{2,5} {2,4}
{3,4} {2,5}
{4,5} {2,6}
{2,4,5} {3,4}
{3,6}
{4,5}
{4,6}
{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
A324762. The case of subsets of {1...n} is
A324738. The strict integer partition version is
A324750. The integer partition version is
A324755. The Heinz number version is
A324760. An infinite version is
A324694.
Cf.
A000720,
A001221,
A001462,
A007097,
A084422,
A085945,
A112798,
A276625,
A279861,
A290689,
A290822,
A304360,
A306844.
-
Table[Length[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]));
((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
A325365
Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 1, 4, 4, 8, 4, 20, 4, 12, 12, 48, 8, 56, 16, 64, 48, 48, 36, 324, 162, 81, 567, 378, 168, 1680, 168, 1848, 264, 264, 264, 2640, 240, 288, 288, 3456, 576, 7488, 1152, 4032, 4032, 2016, 1872, 28080, 9360, 6240, 3360, 6720, 3584, 28672, 6144
Offset: 1
The a(1) = 1 through a(12) = 20 subsets (A = 10, B = 11, C = 12) are the following. The common cardinality of sets in column n is A085089(n).
1 12 13 124 145 1246 1467 12468 12689 1248A 1468B 12468C
134 1346 13468 13689 1289A 148AB 1248AC
1456 14568 15689 1348A 1689B 12689C
14678 16789 1389A 189AB 1289AC
1458A 13468C
1478A 1348AC
1589A 13689C
1789A 1389AC
14568C
1458AC
14678C
1468BC
1478AC
148ABC
15689C
1589AC
16789C
1689BC
1789AC
189ABC
Cf.
A001221,
A001222,
A025487,
A064839,
A085089,
A112798,
A118914,
A124010,
A181819,
A324762,
A325263,
A325365,
A326438,
A326441.
-
prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@Length/@Split[Sort[Array[prisig,n]]]/Count[Array[prisig,n],prisig[n]],{n,30}]
A326439
Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 3, 4, 4, 8, 16, 20, 20, 24, 36, 48, 48, 56, 112, 128, 192, 240, 288, 324, 324, 486, 567, 1134, 1512, 1680, 1680, 1848, 1848, 2112, 2376, 2640, 2640, 2880, 3168, 3456, 6912, 7488, 14976, 16128, 20160, 24192, 26208, 28080, 28080, 37440, 43680
Offset: 0
The a(0) = 1 through a(9) = 8 subsets:
{} {1} {12} {12} {124} {124} {1246} {1246} {12468} {12468}
{13} {134} {134} {1346} {1346} {13468} {12689}
{145} {1456} {1456} {14568} {13468}
{1467} {14678} {13689}
{14568}
{14678}
{15689}
{16789}
Cf.
A001221,
A001222,
A025487,
A064839,
A085089,
A112798,
A118914,
A124010,
A181819,
A324762,
A325263,
A325365,
A326438,
A326441.
-
prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(Length/@Split[Sort[Array[prisig,n]]]),{n,0,30}]
-
a(n)={if(n==0, 1, my(M=Map()); for(i=1, n, my(f=factor(i)[,2], s=sum(k=1, #f, x^f[k]), z); mapput(M, s, if(mapisdefined(M, s, &z), z + 1, 1))); vecprod(Mat(M)[,2]))} \\ Andrew Howroyd, Aug 30 2019
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