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 1-4 of 4 results.

A174063 Triangular array (read by rows) containing the least n integers < 2n such that no integer divides another.

Original entry on oeis.org

1, 2, 3, 2, 3, 5, 2, 3, 5, 7, 4, 5, 6, 7, 9, 4, 5, 6, 7, 9, 11, 4, 5, 6, 7, 9, 11, 13, 4, 6, 7, 9, 10, 11, 13, 15, 4, 6, 7, 9, 10, 11, 13, 15, 17, 4, 6, 7, 9, 10, 11, 13, 15, 17, 19, 4, 6, 9, 10, 11, 13, 14, 15, 17, 19, 21, 4, 6, 9, 10, 11, 13, 14, 15, 17, 19, 21, 23, 4, 6, 9, 10, 11, 13, 14
Offset: 1

Views

Author

David Brown, Mar 07 2010

Keywords

Comments

The first number on each row is 2^k, where k is the greatest integer such that (3^k)/2 < n.

Examples

			1;
2, 3;
2, 3, 5;
2, 3, 5, 7;
4, 5, 6, 7, 9;
4, 5, 6, 7, 9, 11;
4, 5, 6, 7, 9, 11, 13;
4, 6, 7, 9, 10, 11, 13, 15;

Crossrefs

Cf. A174062, A174094.

Programs

  • Mathematica
    noneDivQ[L_] := NoneTrue[Subsets[L, {2}], Divisible[#[[2]], #[[1]]]&];
k1[n_] := For[k = Log[3, 2n]//Ceiling, True, k--, If[(3^k)/2Jean-François Alcover, Sep 25 2020 *)

Extensions

Definition corrected by David Brown, Mar 20 2010

A192298 The number of sets of n positive integers strictly less than 2*n such that no integer in the set divides another.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 6, 6, 10, 14, 13, 26, 34, 24, 48, 72, 60, 120, 168, 168, 264, 396, 312, 624, 816, 816, 1632, 2208
Offset: 1

Views

Author

David Brown, Jun 27 2011

Keywords

Comments

Similar to A174094, but the maximum integer in each set must be strictly less than 2n here.

Examples

			a(1) counts {1};
a(2) counts {2,3};
a(3) counts {2,3,5} and {3,4,5};
a(4) counts {2,3,5,7}, {3,4,5,7}, and {4,5,6,7};
a(5) counts {4,5,6,7,9} and {5,6,7,8,9}.

Crossrefs

Cf. A174094.

Programs

  • Maple
    with(combstruct) ;
A192298nodiv := proc(s) sl := sort(convert(s,list)) ; for i from 1 to nops(sl)-1 do for j from i+1 to nops(sl) do if op(j,sl) mod op(i,sl) = 0 then return false; end if; end do: end do:true ; end proc:
A192298 := proc(n) a := 0 ; it := iterstructs(Subset({seq(i,i=1..2*n-1)},size=n)) :  while not finished(it) do s := nextstruct(it) ; if nops(s) = n then if A192298nodiv(s) then  a := a+1 ; end if; end if; end do: a ; end proc: # R. J. Mathar, Jul 12 2011

A355145 Triangle read by rows: T(n,k) is the number of primitive subsets of {1,...,n} of cardinality k; n>=0, 0<=k<=ceiling(n/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 5, 5, 2, 1, 6, 7, 3, 1, 7, 12, 10, 3, 1, 8, 16, 15, 5, 1, 9, 22, 26, 13, 2, 1, 10, 28, 38, 22, 4, 1, 11, 37, 66, 60, 26, 4, 1, 12, 43, 80, 76, 35, 6, 1, 13, 54, 123, 156, 111, 41, 6, 1, 14, 64, 161, 227, 180, 74, 12
Offset: 0

Views

Author

Marcel K. Goh, Jun 20 2022

Keywords

Comments

A set is primitive if it does not contain distinct i and j such that i divides j.
For n >= 2, the alternating row sums equal -1.

Examples

			Triangle T(n,k) begins:
   n/k 0  1  2  3  4  5  6  7  8  9 10 11 12
    0  1
    1  1  1
    2  1  2
    3  1  3  1
    4  1  4  2
    5  1  5  5  2
    6  1  6  7  3
    7  1  7 12 10  3
    8  1  8 16 15  5
    9  1  9 22 26 13  2
   10  1 10 28 38 22  4
   11  1 11 37 66 60 26  4
   12  1 12 43 80 76 35  6
   ...
For n=6 and k=3 the T(6,3) = 3 primitive sets are {2,3,5}, {3,4,5}, and {4,5,6}.

Links

Crossrefs

Columns k=0..2 give: A000012, A000027, A161664.
Row sums give A051026.
T(2n,n) gives A174094.
T(2n-1,n) gives A192298 for n>=1.
Cf. A087077, A087086.

Formula

Sum_{k=1..ceiling(n/2)} k * T(n,k) = A087077(n). - Alois P. Heinz, Jun 24 2022

A174062 Least possible sum of exactly n positive integers less than 2n such that none of the n integers divides another.

Original entry on oeis.org

1, 5, 10, 17, 31, 42, 55, 75, 92, 111, 139, 162, 187, 233, 262, 293, 337, 372, 409, 461, 502, 545, 615, 662, 711, 779, 832, 887, 963, 1022, 1083, 1181, 1246, 1313, 1405, 1476, 1549, 1649, 1726, 1805, 1951, 2034, 2119, 2235, 2324, 2415, 2539, 2634, 2731, 2885
Offset: 1

Views

Author

David Brown, Mar 06 2010

Keywords

Crossrefs

Cf. A174063, A174094.
Row sums of triangle A174063. [David Brown, Mar 20 2010]

Programs

  • Maple
    f:= proc(N) local i,j,obj,cons;
obj:= add(i*x[i],i=1..2*N-1);
cons:= {seq(seq(x[i]+x[j]<=1, j=2*i..2*N-1, i),i=1..N),
add(x[i],i=1..2*N-1)=N};
Optimization:-Minimize(obj,cons,assume=binary)[1]
end proc:
map(f, [$1..60]); # Robert Israel, May 06 2019
  • Mathematica
    a[n_] := Module[{obj, cons},
obj = Sum[i*x[i], {i, 1, 2n-1}];
cons = Append[Flatten[Table[Table[x[i]+x[j] <= 1, {j, 2i, 2n-1, i}], {i, 1, n}], 1], AllTrue[Array[x, 2n-1], 0 <= # <= 1&] && Sum[x[i], {i, 1, 2n-1}] == n];
Minimize[{obj, cons}, Array[x, 2n-1], Integers][[1]]];
Reap[For[n = 1, n <= 50, n++, Print[n, " ", a[n]]; Sow[a[n]]]][[2, 1]]; (* Jean-François Alcover, May 17 2023, after Robert Israel *)

Extensions

Extended by Ray Chandler, Mar 19 2010
Showing 1-4 of 4 results.

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