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.

Previous Showing 11-20 of 20 results.

A015623 Quadruples of different integers from [ 1,n ] with no common factors between pairs.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 10, 14, 24, 26, 68, 72, 151, 167, 192, 228, 411, 431, 708, 758, 858, 928, 1419, 1475, 1874, 2013, 2353, 2499, 3515, 3571, 4856, 5254, 5747, 6089, 6795, 6979, 9129, 9630, 10437, 10800, 13768, 13978, 17500, 18275, 19210, 20114, 24734
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Partial sums of A185348, Column 4 of triangle A186974.

A186982 Number of 6-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

6, 6, 39, 41, 44, 56, 213, 219, 619, 655, 731, 777, 1853, 1909, 2775, 2961, 3707, 3923, 7803, 7859, 14308, 15876, 17437, 18459, 20865, 21313, 36080, 38103, 41568, 42828, 68688, 69360, 106470, 111456, 117033, 122883, 181854, 185066, 230208, 236610
Offset: 11

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Examples

			a(12) = 6 because there are 6 6-element subsets of {1, 2, ..., 12} having pairwise coprime elements: {1,2,3,5,7,11}, {1,5,7,8,9,11}, {1,4,5,7,9,11}, {1,3,5,7,8,11}, {1,3,4,5,7,11}, {1,2,5,7,9,11}.

Links

Crossrefs

Column 6 of triangle A186974. Partial sums of A186977.

A186983 Number of 7-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

6, 6, 6, 8, 64, 65, 284, 297, 324, 339, 1116, 1144, 1784, 1886, 2449, 2576, 6499, 6527, 14386, 16178, 17683, 18649, 21003, 21395, 42708, 44997, 48966, 50310, 93138, 93852, 163212, 170778, 178896, 187722, 310605, 315687, 408771, 419661, 453255
Offset: 13

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Examples

			a(16) = 8 because there are 8 7-element subsets of {1, 2, ..., 16} having pairwise coprime elements: {1,5,7,9,11,13,16}, {1,5,7,8,9,11,13}, {1,2,3,5,7,11,13}, {1,2,5,7,9,11,13}, {1,3,4,5,7,11,13}, {1,3,5,7,8,11,13}, {1,3,5,7,11,13,16}, {1,4,5,7,9,11,13}.

Links

Crossrefs

Column 7 of triangle A186974. Partial sums of A186978.

A186984 Number of 8-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

8, 8, 73, 75, 79, 81, 420, 428, 719, 750, 1014, 1056, 3632, 3640, 10167, 11603, 12570, 13180, 14698, 14930, 36325, 38118, 41241, 42243, 92553, 93093, 186945, 195339, 203985, 213681, 401403, 407475, 549417, 563277, 608091, 633633, 1106226
Offset: 17

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Examples

			a(17) = 8 because there are 8 8-element subsets of {1, 2, ..., 17} having pairwise coprime elements: {1,2,3,5,7,11,13,17}, {1,2,5,7,9,11,13,17}, {1,3,4,5,7,11,13,17}, {1,3,5,7,8,11,13,17}, {1,3,5,7,11,13,16,17}, {1,4,5,7,9,11,13,17}, {1,5,7,9,11,13,16,17}, {1,5,7,8,9,11,13,17}.

Links

Crossrefs

Column 8 of triangle A186974. Partial sums of A186979.

A186985 Number of 9-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

8, 8, 8, 8, 89, 90, 164, 168, 238, 244, 1300, 1301, 4941, 5734, 6132, 6379, 7000, 7089, 22019, 22975, 24637, 25150, 67393, 67678, 160771, 167602, 174367, 182152, 395833, 401344, 563893, 577192, 621709, 646954, 1280587, 1297318, 1442533, 1478536
Offset: 19

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Examples

			a(19) = 8 because there are 8 9-element subsets of {1, 2, ..., 19} having pairwise coprime elements: {1,3,4,5,7,11,13,17,19}, {1,3,5,7,8,11,13,17,19}, {1,3,5,7,11,13,16,17,19}, {1,2,3,5,7,11,13,17,19}, {1,4,5,7,9,11,13,17,19}, {1,2,5,7,9,11,13,17,19}, {1,5,7,8,9,11,13,17,19}, {1,5,7,9,11,13,16,17,19}.

Links

Crossrefs

Column 9 of triangle A186974. Partial sums of A186980.

A186986 Number of 10-element subsets of {1, 2, ..., n} having pairwise coprime elements.

Original entry on oeis.org

8, 8, 16, 16, 24, 24, 268, 268, 1569, 1857, 1952, 2010, 2156, 2176, 9265, 9596, 10167, 10339, 35489, 35589, 103267, 107303, 111153, 115675, 297827, 301611, 442057, 451649, 484539, 503173, 1150127, 1164141, 1290703, 1320491, 1413885, 1471627
Offset: 23

Views

Author

Alois P. Heinz, Mar 02 2011

Keywords

Examples

			a(24) = 8 because there are 8 10-element subsets of {1, 2, ..., 24} having pairwise coprime elements: {1,2,3,5,7,11,13,17,19,23}, {1,2,5,7,9,11,13,17,19,23}, {1,3,4,5,7,11,13,17,19,23}, {1,3,5,7,8,11,13,17,19,23}, {1,3,5,7,11,13,16,17,19,23}, {1,4,5,7,9,11,13,17,19,23}, {1,5,7,8,9,11,13,17,19,23}, {1,5,7,9,11,13,16,17,19,23}.

Links

Crossrefs

Column 10 of triangle A186974. Partial sums of A186981.

A015698 Number of 5-tuples of different integers from [ 1,n ] with no common factors among pairs.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 3, 6, 6, 32, 33, 105, 114, 128, 157, 385, 400, 831, 886, 1001, 1076, 2004, 2074, 2817, 3022, 3655, 3880, 6379, 6449, 10020, 10978, 12063, 12791, 14417, 14767, 21746, 22977, 25038, 25860, 36660, 37110, 51088, 53449, 56199
Offset: 1

Views

Author

Olivier Gérard

Keywords

Crossrefs

Column 5 of triangle A186974. Partial sums of A186976.

A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 10, 5, 1, 1, 11, 19, 15, 6, 1, 1, 17, 34, 35, 21, 7, 1, 1, 21, 52, 69, 56, 28, 8, 1, 1, 27, 79, 125, 126, 84, 36, 9, 1, 1, 31, 109, 205, 251, 210, 120, 45, 10, 1, 1, 41, 154, 325, 461, 462, 330, 165, 55, 11, 1, 1, 45, 196
Offset: 1

Views

Author

Gus Wiseman, Jan 08 2019

Keywords

Comments

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

Examples

			Triangle begins:
    1
    1    1
    1    3    1
    1    5    4    1
    1    9   10    5    1
    1   11   19   15    6    1
    1   17   34   35   21    7    1
    1   21   52   69   56   28    8    1
    1   27   79  125  126   84   36    9    1
    1   31  109  205  251  210  120   45   10    1
    1   41  154  325  461  462  330  165   55   11    1
    1   45  196  479  786  923  792  495  220   66   12    1
    1   57  262  699 1281 1715 1716 1287  715  286   78   13    1
The T(6,2) = 11 sets are: {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {5,6}. Missing from this list are: {2,4}, {2,6}, {3,6}, {4,6}.

Links

Crossrefs

Row sums are A085945.
Second column is A015614.
Cf. A000837, A186974, A187106, A289508, A289509, A300486, A303139, A320424, A320436.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{k}],GCD@@#==1&]],{n,10},{k,n}]
  • PARI
    T(n,k) = sum(d=1, n\k, moebius(d)*binomial(n\d, k)) \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = Sum_{d=1..floor(n/k)} mu(d)*binomial(floor(n/d), k). - Andrew Howroyd, Jan 19 2023

A319187 Number of pairwise coprime subsets of {1,...,n} of maximum cardinality (A036234).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 24, 24, 24, 24, 24, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 60, 72, 72, 72, 72, 72, 72, 72, 72
Offset: 1

Views

Author

Gus Wiseman, Jan 09 2019

Keywords

Comments

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

Examples

			The a(8) = 3 subsets are {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,7,8}.

Links

Crossrefs

Rightmost terms of A186974 and A320436.
Run lengths are A053707.
Cf. A015614, A036234, A051424, A085945, A186971, A186972, A186994, A276187, A303139, A320423, A320426.
Cf. A000005, A045948.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{PrimePi[n]+1}],CoprimeQ@@#&]],{n,24}] (* see A186974 for a faster program *)
  • PARI
    a(n) = prod(p=1, n, if (isprime(p), logint(n, p), 1)); \\ Michel Marcus, Dec 26 2020

Formula

a(n) = Product_{p prime <= n} floor(log_p(n)).
a(n) = A000005(A045948(n)). - Ridouane Oudra, Sep 02 2019

A355146 Triangle read by rows: T(n,k) is the number of subsets of {1,...,n} of cardinality k in which every pair of elements is coprime; n >= 0, 0 <= k <= A036234(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 11, 8, 2, 1, 7, 17, 19, 10, 2, 1, 8, 21, 25, 14, 3, 1, 9, 27, 37, 24, 6, 1, 10, 31, 42, 26, 6, 1, 11, 41, 73, 68, 32, 6, 1, 12, 45, 79, 72, 33, 6, 1, 13, 57, 124, 151, 105, 39, 6, 1, 14, 63, 138, 167, 114, 41, 6
Offset: 0

Views

Author

Marcel K. Goh, Jun 27 2022

Keywords

Comments

For n >= 1, the alternating row sums equal 0.

Examples

			Triangle T(n,k) begins:
  n/k 0  1  2  3  4  5 6
  0   1
  1   1  1
  2   1  2  1
  3   1  3  3  1
  4   1  4  5  2
  5   1  5  9  7  2
  6   1  6 11  8  2
  7   1  7 17 19 10  2
  8   1  8 21 25 14  3
  9   1  9 27 37 24  6
  10  1 10 31 42 26  6
  11  1 11 41 73 68 32 6
  12  1 12 45 79 72 33 6
  ...
For n=8 and k=5 the T(8,5)=3 sets are {1,2,3,5,7}, {1,3,4,5,7}, and {1,3,5,7,8}.

Links

Crossrefs

Row sums give A084422.
Cf. A000720, A036234, A186974, A320436.
Previous Showing 11-20 of 20 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.