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 1-3 of 3 results.

A186975 Irregular triangle T(n,k), n>=1, 1<=k<=A186971(n), read by rows: T(n,k) is the number of subsets of {1, 2, ..., n} containing n and having <=k pairwise coprime elements.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 3, 4, 1, 5, 10, 12, 1, 3, 4, 1, 7, 18, 26, 28, 1, 5, 11, 15, 16, 1, 7, 19, 29, 32, 1, 5, 10, 12, 1, 11, 42, 84, 110, 116, 1, 5, 11, 15, 16, 1, 13, 58, 137, 209, 242, 248, 1, 7, 21, 37, 46, 48, 1, 9, 30, 55, 69, 72, 1, 9, 33, 69, 98, 110, 112
Offset: 1

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Comments

T(n,k) = T(n,k-1) for k>A186971(n). The triangle contains all values of T up to the last element of each row that is different from its predecessor.

Examples

			T(5,3) = 10 because there are 10 subsets of {1,2,3,4,5} containing n and having <=3 pairwise coprime elements: {5}, {1,5}, {2,5}, {3,5}, {4,5}, {1,2,5}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Triangle T(n,k) begins:
  1;
  1, 2;
  1, 3, 4;
  1, 3, 4;
  1, 5, 10, 12;
  1, 3, 4;
  1, 7, 18, 26, 28;
		

Crossrefs

Columns k=1-9 give: A000012, A039649 for n>1, A186987, A186988, A186989, A186990, A186991, A186992, A186993.
Rightmost elements of rows give A186973.

Programs

  • Maple
    with(numtheory):
    s:= proc(m,r) option remember; mul(`if`(in then 0
        elif k=1 then 1
        elif k=2 and t=n then `if`(n<2, 0, phi(n))
        else c:= 0;
             d:= 2-irem(t, 2);
             for h from 1 to n-1 by d do
               if igcd(t, h)=1 then c:= c +b(s(t*h, h), h, k-1) fi
             od; c
          fi
        end:
    T:= proc(n, k) option remember;
           b(s(n, n), n, k) +`if`(k=0, 0, T(n, k-1))
        end:
    seq(seq(T(n, k), k=1..a(n)), n=1..20);
  • Mathematica
    s[m_, r_] := s[m, r] = Product[If[i < r, i, 1], {i, FactorInteger[m][[All, 1]]}]; a[n_] := a[n] = If[n < 4, n, PrimePi[n]-Length[FactorInteger[n]]+2]; b[t_, n_, k_] := b[t, n, k] = Module[{c, d, h}, Which[k == 0 || k > n, 0, k == 1, 1, k == 2 && t == n, If[n < 2, 0, EulerPhi[n]], True, c = 0; d = 2-Mod[t, 2]; For[h = 1, h <= n-1, h = h+d, If[GCD[t, h] == 1, c = c+b[s[t*h, h], h, k-1] ] ]; c ] ]; t[n_, k_] := t[n, k] = b[s[n, n], n, k]+If[k == 0, 0, t[n, k-1]]; Table[Table[t[n, k], {k, 1, a[n]}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

Formula

T(n,k) = Sum_{i=1..k} A186972(n,i).

A186991 Number of subsets of {1, 2, ..., n} containing n and having <=7 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 248, 48, 72, 112, 720, 64, 1447, 190, 380, 254, 3444, 247, 2795, 701, 2410, 784, 11588, 247, 19472, 4839, 4802, 3175, 7300, 1449, 45641, 6191, 10520, 3908, 82986, 2124, 124554, 15874, 17608
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Crossrefs

Column 7 of triangle A186975. Sum of A186990 and A186978.

A187267 Number of nonempty subsets of {1, 2, ..., n} with <=6 pairwise coprime elements.

Original entry on oeis.org

1, 3, 7, 11, 23, 27, 55, 71, 103, 115, 231, 247, 489, 537, 609, 719, 1383, 1446, 2674, 2851, 3204, 3443, 6110, 6329, 8484, 9083, 10930, 11587, 19252, 19471, 31084, 34131, 37428, 39637, 44583, 45640, 69968, 73870, 80421, 82985
Offset: 1

Views

Author

Alois P. Heinz, Mar 07 2011

Keywords

Crossrefs

Column 6 of triangle A187262. First differences are A186990.

Formula

a(n) = A187262(n,6).
Showing 1-3 of 3 results.