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.

A343653 Number of non-singleton pairwise coprime nonempty sets of divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Apr 25 2021

Keywords

Comments

First differs from A066620 at a(210) = 36, A066620(210) = 35.

Examples

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {2,3}    {2,3}  {2,3}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,5}    {2,9}  {2,5}    {2,9}  {3,4}
                {3,8}  {3,5}    {3,4}  {3,4}    {3,4}  {3,8}
                       {5,6}    {4,9}  {3,5}    {3,8}  {3,16}
                       {2,15}          {4,5}    {4,9}  {3,32}
                       {3,10}          {5,6}    {8,9}
                       {2,3,5}         {2,15}
                                       {3,10}
                                       {3,20}
                                       {4,15}
                                       {5,12}
                                       {2,3,5}
                                       {3,4,5}

Crossrefs

The case of pairs is A089233.
The version with 1's, empty sets, and singletons is A225520.
The version for subsets of {1..n} is A320426.
The version for strict partitions is A337485.
The version for compositions is A337697.
The version for prime indices is A337984.
The maximal case with 1's is A343652.
The version with empty sets is a(n) + 1.
The version with singletons is A343654(n) - 1.
The version with empty sets and singletons is A343654.
The version with 1's is A343655.
The maximal case is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A343659 counts maximal pairwise coprime subsets of {1..n}.
Cf. A007359, A067824, A074206, A076078, A084422, A187106, A285572, A324837, A326675, A327516, A338315.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Rest[Divisors[n]]],CoprimeQ@@#&]],{n,100}]

A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 265, 275, 279, 287, 291, 295, 297, 309, 323, 327, 329, 335, 341, 355, 363, 369
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2020

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also Heinz numbers of integer partitions with no 1's whose distinct parts are pairwise coprime (A338315). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
     15: {2,3}      135: {2,2,2,3}    215: {3,14}
     33: {2,5}      141: {2,15}       217: {4,11}
     35: {3,4}      143: {5,6}        219: {2,21}
     45: {2,2,3}    145: {3,10}       221: {6,7}
     51: {2,7}      153: {2,2,7}      225: {2,2,3,3}
     55: {3,5}      155: {3,11}       245: {3,4,4}
     69: {2,9}      161: {4,9}        249: {2,23}
     75: {2,3,3}    165: {2,3,5}      253: {5,9}
     77: {4,5}      175: {3,3,4}      255: {2,3,7}
     85: {3,7}      177: {2,17}       265: {3,16}
     93: {2,11}     187: {5,7}        275: {3,3,5}
     95: {3,8}      201: {2,19}       279: {2,2,11}
     99: {2,2,5}    205: {3,13}       287: {4,13}
    119: {4,7}      207: {2,2,9}      291: {2,25}
    123: {2,13}     209: {5,8}        295: {3,17}

Crossrefs

A304711 is the not necessarily odd version, with squarefree case A302797.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A337984 is the squarefree case.
A338315 counts the partitions with these Heinz numbers.
A338316 considers singletons coprime.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's, with Heinz numbers A302568.
A304709 counts partitions whose distinct parts are pairwise coprime.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
A318717 counts pairwise non-coprime strict partitions, with Heinz numbers A318719.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A289508, A289509, A302569, A303282, A328867, A337485.

Programs

  • Mathematica
    Select[Range[1,100,2],CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 95, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 131, 135, 137, 139, 141, 143, 145, 149, 151
Offset: 1

Views

Author

Gus Wiseman, Oct 24 2020

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. a(n) gives the n-th Heinz number of an integer partition with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime (A338317).

Examples

			The sequence of terms together with their prime indices begins:
      1: {}          33: {2,5}       71: {20}
      3: {2}         35: {3,4}       73: {21}
      5: {3}         37: {12}        75: {2,3,3}
      7: {4}         41: {13}        77: {4,5}
      9: {2,2}       43: {14}        79: {22}
     11: {5}         45: {2,2,3}     81: {2,2,2,2}
     13: {6}         47: {15}        83: {23}
     15: {2,3}       49: {4,4}       85: {3,7}
     17: {7}         51: {2,7}       89: {24}
     19: {8}         53: {16}        93: {2,11}
     23: {9}         55: {3,5}       95: {3,8}
     25: {3,3}       59: {17}        97: {25}
     27: {2,2,2}     61: {18}        99: {2,2,5}
     29: {10}        67: {19}       101: {26}
     31: {11}        69: {2,9}      103: {27}

Crossrefs

A338315 does not consider singletons coprime, with Heinz numbers A337987.
A338317 counts the partitions with these Heinz numbers.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A007359 counts singleton or pairwise coprime partitions with no 1's, with Heinz numbers A302568.
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
A302797 lists squarefree numbers whose distinct parts are pairwise coprime.
A304709 counts partitions whose distinct parts are pairwise coprime, with Heinz numbers A304711.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337485 counts pairwise coprime partitions with no 1's, with Heinz numbers A337984.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A056239, A112798, A220377, A289509, A302569, A303282, A318719, A328673, A328867.

Programs

  • Mathematica
    Select[Range[1,100,2],#==1||PrimePowerQ[#]||CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376
Offset: 0

Views

Author

Gus Wiseman, Oct 24 2020

Keywords

Examples

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222

Crossrefs

A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Cf. A051424, A289508, A302569, A303140, A337694.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

Formula

The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Showing 1-4 of 4 results.

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