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 1-9 of 9 results.

A187262 Irregular triangle T(n,k), n>=1, 1<=k<=A036234(n), read by rows: T(n,k) is the number of nonempty subsets of {1, 2, ..., n} having <=k pairwise coprime elements.

Original entry on oeis.org

1, 2, 3, 3, 6, 7, 4, 9, 11, 5, 14, 21, 23, 6, 17, 25, 27, 7, 24, 43, 53, 55, 8, 29, 54, 68, 71, 9, 36, 73, 97, 103, 10, 41, 83, 109, 115, 11, 52, 125, 193, 225, 231, 12, 57, 136, 208, 241, 247, 13, 70, 194, 345, 450, 489, 495, 14, 77, 215, 382, 496, 537, 543
Offset: 1

Views

Author

Alois P. Heinz, Mar 07 2011

Keywords

Comments

T(n,k) = T(n,k-1) for k>A036234(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) = 21 because there are 21 nonempty subsets of {1,2,3,4,5} having <=3 pairwise coprime elements: {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,5}, {3,4,5}.
Irregular Triangle T(n,k) begins:
  1;
  2,  3;
  3,  6,  7;
  4,  9, 11;
  5, 14, 21, 23;
  6, 17, 25, 27;
  7, 24, 43, 53, 55;

Links

Crossrefs

Columns k=1-10 give: A000027, A187263, A187264, A187265, A187266, A187267, A187268, A187269, A187270, A187271.
Rightmost elements of rows give A187106.
Cf. A036234, A186972, A186974, A186975.

Formula

T(n,k) = Sum_{i=1..n,j=1..k} A186972(i,j).
T(n,k) = Sum_{j=1..k} A186974(n,j).
T(n,k) = Sum_{i=1..n} A186975(i,k).

A186973 Number of subsets of {1, 2, ..., n} containing n and having pairwise coprime elements; also row sums of A186972.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 248, 48, 72, 112, 728, 64, 1520, 192, 384, 256, 3872, 256, 3168, 736, 2752, 832, 15488, 256, 31232, 7424, 6272, 4096, 9600, 1792, 91648, 9344, 16000, 5632, 214272, 3072, 431616, 37376, 38912, 43008, 982528
Offset: 1

Views

Author

Alois P. Heinz, Mar 01 2011

Keywords

Examples

			a(6) = 4 because there are 4 subsets of {1,2,3,4,5,6} containing 6 and having pairwise coprime elements: {6}, {1,6}, {5,6}, {1,5,6}.

Links

Crossrefs

Cf. A186971, A186972, A186994. Rightmost elements in rows of triangle A186975.

Programs

  • Maple
    with(numtheory):
s:= proc(m, r) option remember; mul(`if`(i mul(ilog[j](n), j={ithprime(i)$i=1..pi(n)} minus factorset(n)):
b:= proc(t, n, k) option remember; local c, d, h;
      if k=0 or k>n 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
         end; c
      fi
    end:
a:= n-> h(n) + add(b(s(n, n), n, k), k=1..g(n)-1):
seq(a(n), n=1..50);
  • Mathematica
    s[m_, r_] := s[m, r] = Product[If[in, 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]; Table[Sum[t[n, k], {k, 1, a[n]}], {n, 1, 50}] (* Jean-François Alcover, Dec 04 2014, after Alois P. Heinz *)

A186987 Number of subsets of {1, 2, ..., n} containing n and having <=3 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 10, 4, 18, 11, 19, 10, 42, 11, 58, 21, 30, 33, 96, 22, 120, 36, 62, 48, 172, 37, 147, 69, 128, 70, 270, 37, 308, 123, 158, 117, 208, 75, 432, 147, 218, 119, 530, 78, 584, 186, 228, 212, 696, 133, 594, 191, 380, 256, 882, 166, 547
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Examples

			a(6) = 4 because there are 4 subsets of {1,2,3,4,5,6} containing 6 and having <=3 pairwise coprime elements: {6}, {1,6}, {5,6}, {1,5,6}.

Links

Crossrefs

Column 3 of triangle A186975. Sum of A039649 and A185953 for n>1.

A186988 Number of subsets of {1, 2, ..., n} containing n and having <=4 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 26, 15, 29, 12, 84, 15, 137, 37, 55, 69, 279, 42, 397, 86, 162, 118, 663, 93, 546, 208, 468, 216, 1286, 93, 1593, 521, 651, 459, 914, 259, 2582, 648, 1025, 482, 3498, 288, 4106, 961, 1163, 1116, 5316, 641, 4326, 1033
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Examples

			a(5) = 12 because there are 12 subsets of {1,2,3,4,5} containing 5 and having <=4 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}, {1,2,3,5}, {1,3,4,5}.

Links

Crossrefs

Column 4 of triangle A186975. Sum of A186987 and A185348.

A186989 Number of subsets of {1, 2, ..., n} containing n and having <=5 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 110, 16, 209, 46, 69, 98, 507, 57, 828, 141, 277, 193, 1591, 163, 1289, 413, 1101, 441, 3785, 163, 5164, 1479, 1736, 1187, 2540, 609, 9561, 1879, 3086, 1304, 14298, 738, 18084, 3322, 3913, 3888, 25430
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Examples

			a(10) = 12 because there are 12 subsets of {1, 2, ..., 10} containing 10 and having <=5 pairwise coprime elements: {10}, {1,10}, {3,10}, {7,10}, {9,10}, {1,3,10}, {1,7,10}, {1,9,10}, {3,7,10}, {7,9,10}, {1,3,7,10}, {1,7,9,10}. There is no subset with exactly 5 elements here.

Links

Crossrefs

Column 5 of triangle A186975. Sum of A186988 and A186976.

A186990 Number of subsets of {1, 2, ..., n} containing n and having <=6 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 242, 48, 72, 110, 664, 63, 1228, 177, 353, 239, 2667, 219, 2155, 599, 1847, 657, 7665, 219, 11613, 3047, 3297, 2209, 4946, 1057, 24328, 3902, 6551, 2564, 40158, 1410, 55194, 8308, 9490
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Links

Crossrefs

Column 6 of triangle A186975. Sum of A186989 and A186977.

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

Links

Crossrefs

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

A186992 Number of subsets of {1, 2, ..., n} containing n and having <=8 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 248, 48, 72, 112, 728, 64, 1512, 192, 384, 256, 3783, 255, 3086, 732, 2674, 826, 14164, 255, 25999, 6275, 5769, 3785, 8818, 1681, 67036, 7984, 13643, 4910, 133296, 2664, 218406, 24268, 26254
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Links

Crossrefs

Column 8 of triangle A186975. Sum of A186991 and A186979.

A186993 Number of subsets of {1, 2, ..., n} containing n and having <=9 pairwise coprime elements.

Original entry on oeis.org

1, 2, 4, 4, 12, 4, 28, 16, 32, 12, 116, 16, 248, 48, 72, 112, 728, 64, 1520, 192, 384, 256, 3864, 256, 3160, 736, 2744, 832, 15220, 256, 29639, 7068, 6167, 4032, 9439, 1770, 81966, 8940, 15305, 5423, 175539, 2949, 311499, 31099, 33019
Offset: 1

Views

Author

Alois P. Heinz, Mar 03 2011

Keywords

Links

Crossrefs

Column 9 of triangle A186975. Sum of A186992 and A186980.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.