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.

A084422 Number of subsets of integers 1 through n (including the empty set) containing no pair of integers that share a common factor.

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 28, 56, 72, 104, 116, 232, 248, 496, 544, 616, 728, 1456, 1520, 3040, 3232, 3616, 3872, 7744, 8000, 11168, 11904, 14656, 15488, 30976, 31232, 62464, 69888, 76160, 80256, 89856, 91648, 183296, 192640, 208640, 214272, 428544
Offset: 0

Views

Author

Matthew Vandermast, Jun 26 2003

Keywords

Comments

Also the number of subsets of {1,...,n} whose product of elements is equal to the least common multiple of elements. - Michel Marcus, Mar 27 2016

Examples

			Exactly 4 of the 2^4=16 subsets of the integers from 1 through 4 contain a pair of integers that share a common factor; these are {2,4}, {1,2,4}, {2,3,4} and {1,2,3,4}. The other 12 subsets do not; hence a(4)=12.

References

  • Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication. [Apparently unpublished as of 2016]

Links

Crossrefs

Cf. A051026 gives the number of primitive subsets. A087080 gives the number of elements in coprime subsets. A087081 gives the sum of the elements in coprime subsets.
Cf. A036234, A186974.

Programs

  • Mathematica
    Prepend[Table[Length@ Select[Rest@ Subsets@ Range@ n, Times @@ # == LCM @@ # &], {n, 22}] + 1, 1] (* Michael De Vlieger, Mar 27 2016 *)
  • PARI
    a(n)=nb = 0; S = vector(n, k, k); for (i = 0, 2^n - 1, ss = vecextract(S, i); if (prod(k=1, #ss, ss[k]) == lcm(ss), nb++);); nb; \\ Michel Marcus, Mar 27 2016
  • PARI
    a(n,k=1)=if(n<2, return(n+1)); if(gcd(k,n)==1, a(n-1,n*k)) + a(n-1,k) \\ Charles R Greathouse IV, Aug 24 2016

Formula

a(n) = 1 + Sum_{k=1..A036234(n)} A186974(n,k) if n>0; a(0) = 1.

Extensions

More terms from Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 12 2003

A087077 Total number of elements in all primitive subsets of the integers 1 to n.

Original entry on oeis.org

0, 1, 2, 5, 8, 21, 29, 73, 105, 193, 288, 677, 853, 1957, 2961, 4913, 6809, 15145, 19605, 43105, 57889, 98849, 151457, 327505, 397825, 784945, 1201189, 2009229, 2772729, 5901185, 7364945, 15609825, 21206049, 36440033, 55602033, 105010513, 127336513, 267374561
Offset: 0

Views

Author

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 10 2003

Keywords

Comments

A primitive set has no element that divides another element in the same set.

Examples

			a(4)=8 since the primitive subsets of (1,2,3,4) are ( ) (1) (2) (3) (4) (2,3) (3,4) and these contain eight elements

References

  • R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, (1994).

Links

Crossrefs

A051026 gives the number of primitive subsets. A087078 gives the sum of the elements of the primitive subsets. A087080 gives the number elements in the coprime subsets.
Cf. A355145.

Formula

a(n) = Sum_{k=1..ceiling(n/2)} k * A355145(n,k). - Alois P. Heinz, Jun 27 2022

Extensions

Terms a(34)-a(37) from Fausto A. C. Cariboni, Feb 02 2022

A087081 Sum of the elements in the coprime subsets of the integers 1 to n.

Original entry on oeis.org

0, 1, 6, 24, 48, 156, 192, 580, 836, 1444, 1660, 4596, 4980, 13184, 14768, 17308, 21756, 55888, 58768, 146416, 157552, 181008, 196304, 481664, 500096, 765648, 825152, 1073920, 1148288, 2745728, 2768768, 6505728, 7453952, 8233792, 8736960, 9984832, 10208064
Offset: 0

Views

Author

Alan Sutcliffe (alansut(AT)ntlworld.com), Aug 12 2003

Keywords

Comments

A coprime set of integers has (m,n)=1 for each pair of integers in the set.

Examples

			a(4)=48 since the 12 coprime subsets of (1,2,3,4) are ( ) (1) (2) (3) (4) (1,2) (1,3) (1,4) (2,3) (3,4) (1,2,3) (1,3 4) and the sum of the elements is 48.

References

  • Alan Sutcliffe, Divisors and Common Factors in Sets of Integers, awaiting publication.

Links

Crossrefs

A087078 gives the sum of the elements in the primitive subsets. A084422 gives the number coprime subsets. A087080 gives the number of elements in coprime subsets.

Extensions

Terms a(35) and beyond from Fausto A. C. Cariboni, Oct 20 2020
Showing 1-3 of 3 results.

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