A326489
Number of product-free subsets of {1..n}.
Original entry on oeis.org
1, 1, 2, 4, 6, 12, 22, 44, 88, 136, 252, 504, 896, 1792, 3392, 6352, 9720, 19440, 35664, 71328, 129952, 247232, 477664, 955328, 1700416, 2657280, 5184000, 10368000, 19407360, 38814720, 68868352, 137736704, 260693504, 505830400, 999641600, 1882820608, 2807196672
Offset: 0
The a(0) = 1 through a(6) = 22 subsets:
{} {} {} {} {} {} {}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{2,3} {4} {4} {4}
{2,3} {5} {5}
{3,4} {2,3} {6}
{2,5} {2,3}
{3,4} {2,5}
{3,5} {2,6}
{4,5} {3,4}
{2,3,5} {3,5}
{3,4,5} {3,6}
{4,5}
{4,6}
{5,6}
{2,3,5}
{2,5,6}
{3,4,5}
{3,4,6}
{3,5,6}
{4,5,6}
{3,4,5,6}
Product-closed subsets are
A326076.
Subsets containing no products are
A326114.
Subsets containing no products of distinct elements are
A326117.
Subsets containing no quotients are
A327591.
Maximal product-free subsets are
A326496.
-
Table[Length[Select[Subsets[Range[n]],Intersection[#,Times@@@Tuples[#,2]]=={}&]],{n,10}]
a(0)=1 prepended to data, example and b-file by
Peter Kagey, Sep 18 2019
A326496
Number of maximal product-free subsets of {1..n}.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 6, 6, 9, 9, 15, 17, 30, 30, 46, 46, 51, 61, 103, 103, 129, 158, 282, 282, 322, 322, 553, 553, 615, 689, 1247, 1365, 1870, 1870, 3566, 3758, 5244, 5244, 8677, 8677, 9807, 12147, 23351, 23351, 27469, 31694, 45718, 47186, 54594, 54594, 95382, 108198
Offset: 0
The a(2) = 1 through a(10) = 6 subsets (A = 10):
{2} {23} {23} {235} {235} {2357} {23578} {23578} {23578}
{34} {345} {256} {2567} {25678} {256789} {2378A}
{3456} {34567} {345678} {345678} {256789}
{456789} {26789A}
{345678A}
{456789A}
- Fausto A. C. Cariboni, Table of n, a(n) for n = 0..85
- P. J. Cameron and P. Erdős, On the number of integers with various properties, in R. A. Mullin, ed., Number Theory: Proc. First Conf. of Canad. Number Theory Assoc. Conf., Banff, De Gruyter, Berlin, 1990, pp. 61-79.
- Andrew Howroyd, PARI Program
Subsets without products of distinct elements are
A326117.
Maximal sum-free subsets are
A121269.
Maximal sum-free and product-free subsets are
A326497.
Maximal subsets without products of distinct elements are
A325710.
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fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Times@@@Tuples[#,2]]=={}&]]],{n,0,10}]
-
\\ See link for program file.
for(n=0, 30, print1(A326496(n), ", ")) \\ Andrew Howroyd, Aug 30 2019
A326491
Number of maximal subsets of {1..n} containing no differences or quotients of pairs of distinct elements.
Original entry on oeis.org
1, 1, 2, 2, 3, 4, 5, 7, 9, 10, 16, 22, 27, 39, 52, 70, 90, 120, 150, 198, 262, 357, 448, 602, 782, 1004, 1294, 1715, 2229, 2932, 3698, 4844, 6259, 8188, 10274, 13446, 16895, 21954, 27470, 35843, 45411, 58949, 73940, 95200, 120594, 154511, 192996, 247967, 312643
Offset: 0
The a(1) = 1 through a(9) = 10 subsets:
{1} {1} {1} {1} {1} {1} {1} {1} {1}
{2} {2,3} {2,3} {2,3} {2,3} {2,3,7} {2,5,6} {2,6,7}
{3,4} {2,5} {2,5,6} {2,5,6} {2,5,8} {3,4,5}
{3,4,5} {3,4,5} {2,6,7} {2,6,7} {3,5,7}
{4,5,6} {3,4,5} {3,4,5} {2,3,7,8}
{3,5,7} {3,5,7} {2,5,6,9}
{4,5,6,7} {2,3,7,8} {2,5,8,9}
{4,5,6,7} {4,5,6,7}
{5,6,7,8} {4,6,7,9}
{5,6,7,8,9}
Subsets without differences or quotients are
A326490.
Subsets with differences and quotients are
A326494.
Maximal subsets without differences are
A121269Maximal subsets without quotients are
A326492.
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fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Union[Divide@@@Reverse/@Subsets[#,{2}],Subtract@@@Reverse/@Subsets[#,{2}]]]=={}&]]],{n,0,10}]
A326490
Number of subsets of {1..n} containing no differences or quotients of pairs of distinct elements.
Original entry on oeis.org
1, 2, 3, 5, 7, 12, 18, 31, 46, 72, 102, 172, 259, 428, 607, 989, 1329, 2142, 3117, 4953, 6956, 11032, 15321, 23979, 33380, 48699, 66849, 104853, 144712, 220758, 304133, 461580, 636556, 973843, 1316513, 1958828, 2585433, 3882843, 5237093, 7884277, 10555739, 15729293
Offset: 0
The a(0) = 1 through a(6) = 18 subsets:
{} {} {} {} {} {} {}
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{2,3} {4} {4} {4}
{2,3} {5} {5}
{3,4} {2,3} {6}
{2,5} {2,3}
{3,4} {2,5}
{3,5} {2,6}
{4,5} {3,4}
{3,4,5} {3,5}
{4,5}
{4,6}
{5,6}
{2,5,6}
{3,4,5}
{4,5,6}
Subsets without difference are
A007865.
Maximal subsets without differences or quotients are
A326491.
Subsets without quotients are
A327591.
Subsets with differences and quotients are
A326494.
-
Table[Length[Select[Subsets[Range[n]],Intersection[#,Union[Divide@@@Reverse/@Subsets[#,{2}],Subtract@@@Reverse/@Subsets[#,{2}]]]=={}&]],{n,0,10}]
-
a(n)={
my(recurse(k, b)=
if(k > n, 1,
my(t = self()(k + 1, b));
for(i=1, k\2, if(bittest(b,i) && (bittest(b,k-i) || (!(k%i) && bittest(b,k/i))), return(t)));
t += self()(k + 1, b + (1<Andrew Howroyd, Aug 25 2019
A326492
Number of maximal subsets of {1..n} containing no quotients of pairs of distinct elements.
Original entry on oeis.org
1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 7, 7, 10, 10, 16, 18, 31, 31, 47, 47, 52, 62, 104, 104, 130, 159, 283, 283, 323, 323, 554, 554, 616, 690, 1248, 1366, 1871, 1871, 3567, 3759, 5245, 5245, 8678, 8678, 9808, 12148, 23352, 23352, 27470, 31695, 45719, 47187, 54595, 54595, 95383, 108199
Offset: 0
The a(0) = 1 through a(9) = 5 subsets:
{} {1} {1} {1} {1} {1} {1} {1} {1} {1}
{2} {23} {23} {235} {235} {2357} {23578} {23578}
{34} {345} {256} {2567} {25678} {256789}
{3456} {34567} {345678} {345678}
{456789}
Subsets with quotients are
A326023.
Subsets with quotients > 1 are
A326079.
Subsets without quotients are
A327591.
Maximal subsets without differences or quotients are
A326491.
Maximal subsets without quotients (or products) are
A326496.
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fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],Intersection[#,Divide@@@Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]]=={}&]]],{n,0,10}]
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