cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A187489 Irregular triangle T(n,k), n>=0, 0<=k<=A068063(n), read by rows: T(n,k) is the number of k-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 1, 6, 7, 1, 7, 12, 1, 1, 8, 16, 2, 1, 9, 22, 6, 1, 10, 28, 12, 1, 1, 11, 37, 22, 2, 1, 12, 43, 31, 3, 1, 13, 54, 49, 6, 1, 14, 64, 70, 10, 1, 15, 75, 99, 21, 1, 16, 86, 128, 32, 1, 17, 101, 176, 49, 1, 18, 113, 216, 65, 1, 19, 130, 284, 101
Offset: 0

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.
T(n,k) = 0 for k>A068063(n). The triangle contains all positive values of T.

Examples

			T(5,2) = 5, because there are 5 2-element nondividing subsets of {1,2,3,4,5}: {2,3}, {2,5}, {3,4}, {3,5}, {4,5}.  T(7,3) = 1: {4,6,7}.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2;
  1, 3, 1;
  1, 4, 2;
  1, 5, 5;
  1, 6, 7;
  1, 7, 12, 1;
  ...

Links

Crossrefs

Columns k=0-8 give: A000012, A000027, A161664, A187490, A187491, A187492, A187493, A187494, A187550.
Row sums give: A051014.
Cf. A068063.

A051014 Number of nondividing sets on {1,2,...,n}.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 14, 21, 27, 38, 52, 73, 90, 123, 159, 211, 263, 344, 413, 535, 658, 832, 1026, 1276, 1499, 1846, 2226, 2708, 3229, 3912, 4592, 5541, 6495, 7795, 9207, 10908, 12547, 14852, 17358, 20493, 23709, 27744, 31921, 37250, 43013, 49936, 57319, 66318
Offset: 0

Views

Author

Eric W. Weisstein

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(5) = 11 because there are 11 nondividing subsets of {1,2,3,4,5}: {}, {1}, {2}, {3}, {4}, {5}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}.
a(7) = 21: {}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {2,3}, {2,5}, {2,7}, {3,4}, {3,5}, {3,7}, {4,5}, {4,6}, {4,7}, {5,6}, {5,7}, {6,7}, {4,6,7}.

Links

Crossrefs

Row sums of A187489. Cf. A068063.

Programs

  • Maple
    sums:= proc(s) option remember; local i, m;
          m:= max(s[]);
         `if`(m<1, {}, {m, seq([i,i+m][], i=sums(s minus {m}))})
       end:
b:= proc(i,s) option remember; local j, ok, t, si;
      if i<2 then 1
    else si:= s union {i};
         ok:= true;
         for j in sums(si) while ok do
           for t in si while ok do
             if irem(j, t)=0 and t<>j then ok:= false fi
           od
         od;
         b(i-1,s) +`if`(ok, b(i-1, si), 0)
      fi
    end:
a:= n-> `if`(n=0, 1, 1+b(n, {})):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 08 2011
  • Mathematica
    sums[s_] := sums[s] = Module[{m=Max[s]},
If[m<1, {},
  Join[{m},
  Sequence@@Table[{i,i+m}, {i,sums[DeleteCases[s,m]]}]]]
];
b[i_,s_] := b[i,s] = Module[{ ok,si,sij,sik},
If[ i<2, 1, si = Union[s,{i}];
ok = True;
For[j=1, j <= Length[sums[si]] && ok, j++,
  sij = sums[si][[j]];
  For[k=1, k <= Length[si] && ok, k++,
    If[Divisible[sij,sik=si[[k]]]&&sij!=sik,ok=False]]];
    b[i-1, s] + If[ok, b[i-1,si],0]
  ]
];
a[n_] := a[n] = If[n==0, 1, 1+b[n, {}]];
Table[ Print[ a[n] ]; a[n], {n,0,47}]
(* Jean-François Alcover, Oct 10 2012, after Alois P. Heinz *)

Extensions

More terms from David Wasserman, Feb 15 2002
a(41)-a(47) from Alois P. Heinz, Mar 08 2011

A187490 Number of 3-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 6, 12, 22, 31, 49, 70, 99, 128, 176, 216, 284, 343, 423, 515, 633, 722, 860, 1007, 1173, 1333, 1552, 1729, 1989, 2223, 2502, 2809, 3138, 3416, 3819, 4226, 4658, 5049, 5570, 6016, 6601, 7146, 7719, 8371, 9100, 9686, 10461, 11208, 12039
Offset: 7

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(7) = 1 because there is one 3-element nondividing subset of {1,2,3,4,5,6,7}: {4,6,7}.
a(9) = 6: {4,6,7}, {4,6,9}, {5,6,8}, {5,8,9}, {6,7,9}, {6,8,9}.

Links

Crossrefs

Column 3 of triangle A187489. Cf. A068063.

A187491 Number of 4-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 6, 10, 21, 32, 49, 65, 101, 150, 224, 305, 413, 525, 707, 908, 1174, 1479, 1871, 2269, 2826, 3396, 4138, 4967, 5991, 6917, 8244, 9673, 11328, 12958, 15091, 17112, 19771, 22468, 25485, 28870, 32861, 36298, 40969, 45615, 51015
Offset: 10

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(10) = 1 because there is one 4-element nondividing subset of {1,2,...,10}: {6,7,9,10}.

Links

Crossrefs

Column 4 of triangle A187489. Cf. A068063.

A187492 Number of 5-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

2, 4, 6, 11, 15, 24, 50, 83, 127, 209, 310, 431, 679, 921, 1229, 1624, 2145, 2770, 3752, 4866, 6141, 7753, 9679, 12005, 15027, 18134, 22045, 26368, 31712, 37763, 45569, 53810, 63393, 73560, 86496, 100071, 117234, 134623, 155465, 176876
Offset: 21

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(21) = 2 because there are two 5-element nondividing subsets of {1,2,...,21}: {12,16,18,19,21}, {12,14,18,20,21}.

Links

Crossrefs

Column 5 of triangle A187489. Cf. A068063.

A187493 Number of 6-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 3, 4, 7, 15, 27, 45, 55, 85, 133, 199, 262, 378, 534, 803, 999, 1319, 1742, 2309, 3007, 4020, 5166, 6565, 7950, 10380, 12882, 16533, 19664, 24099, 30912, 37550, 44092, 54465, 65117, 79616, 94144, 111780, 132592, 159228, 187506, 219949, 256514
Offset: 31

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(31) = 1 because there is one 6-element nondividing subset of {1,2,...,31}: {16,19,23,24,28,31}.

Links

Crossrefs

Column 6 of triangle A187489. Cf. A068063.

A187494 Number of 7-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 9, 12, 21, 27, 41, 46, 74, 99, 137, 153, 203, 307, 414, 464, 612, 788, 1126, 1292, 1645, 2039, 2614, 3291, 4120, 5127, 6356, 7180, 9786
Offset: 43

Views

Author

Alois P. Heinz, Mar 10 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(43) = 1 because there is one 7-element nondividing subset of {1,2,...,43}: {24,28,31,35,36,40,43}.

Links

Crossrefs

Column 7 of triangle A187489. Cf. A068063.

A187550 Number of 8-element nondividing subsets of {1, 2, ..., n}.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 12, 15, 19, 24, 35, 46, 60, 83, 102, 135
Offset: 65

Views

Author

Alois P. Heinz, Mar 11 2011

Keywords

Comments

A set is called nondividing if no element divides the sum of any nonempty subset of the other elements.

Examples

			a(65) = 2 because there are two 8-element nondividing subsets of {1,2,...,65}: {36,40,48,49,53,61,64,65}, {30,44,45,49,50,59,64,65}.

Links

Crossrefs

Column 8 of triangle A187489. Cf. A068063.
Showing 1-8 of 8 results.

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