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.

A305713 Number of strict integer partitions of n into pairwise coprime parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 10, 9, 12, 16, 18, 20, 21, 20, 23, 31, 36, 36, 37, 39, 44, 54, 64, 68, 65, 63, 74, 85, 99, 112, 106, 105, 121, 144, 164, 173, 166, 161, 178, 221, 252, 254, 254, 254, 272, 327, 372, 375, 368, 376, 405, 475, 552, 568, 536
Offset: 1

Views

Author

Gus Wiseman, Sep 01 2018

Keywords

Examples

			The a(13) = 9 strict partitions are (7,6), (8,5), (9,4), (10,3), (11,2), (12,1), (7,5,1), (5,4,3,1), (7,3,2,1).

Links

Crossrefs

Cf. A000009, A051424, A078374, A302569, A302696, A302797, A303282, A303283, A318717, A318719.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&CoprimeQ@@#&]],{n,30}]

A318717 Number of strict integer partitions of n in which no two parts are relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 2, 32, 13, 38, 7, 57, 2, 54, 19, 68, 3, 95, 3, 90, 33, 104, 3, 148, 7, 149, 40, 166, 5, 230, 17, 226, 56, 256, 6, 360, 9, 340, 84, 390, 25, 527, 11, 513, 109
Offset: 0

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(20) = 11 partitions:
  (20),
  (12,8), (14,6), (15,5), (16,4), (18,2),
  (10,6,4), (10,8,2), (12,6,2), (14,4,2),
  (8,6,4,2).

Links

Crossrefs

Cf. A000009, A007359, A007360, A051185, A302569, A302797, A303140, A303280, A303283, A305713, A305843, A305854, A305713, A318715, A318719.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]]&]],{n,30}]

Extensions

a(51)-a(69) from Alois P. Heinz, Sep 02 2018

A318715 Number of strict integer partitions of n with relatively prime parts in which no two parts are relatively prime.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 1, 0, 4, 0, 3, 0, 1, 0, 5, 0, 8, 0, 2, 0, 5, 0, 10, 0, 4, 0, 13, 0, 15, 0, 3, 1, 13, 0, 19, 0, 9, 1, 24, 0, 20
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Examples

			The a(67) = 10 strict integer partitions are
  (45,12,10) (42,15,10) (40,15,12) (33,22,12) (28,21,18)
  (36,15,10,6) (30,15,12,10) (28,21,12,6) (24,18,15,10)
  (24,15,12,10,6).

Links

Crossrefs

Cf. A007359, A051185, A078374, A303140, A303282, A303283, A305713, A305843, A305854, A318716, A318717, A318720.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==1,And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]]&]],{n,50}]

Extensions

a(71)-a(85) from Robert Price, Sep 08 2018

A318719 Heinz numbers of strict integer partitions in which no two parts are relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 197
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Crossrefs

Cf. A302797, A303140, A303282, A303283, A305713, A305843, A318717, A318715, A318716, A318717.

Programs

  • Mathematica
    Select[Range[200],And[SquareFreeQ[#],And@@(GCD[##]>1&)@@@Select[Tuples[PrimePi/@FactorInteger[#][[All,1]],2],Less@@#&]]&]

A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 128, 129, 131, 133, 137, 139, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Oct 30 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A partition with no two distinct parts relatively prime is said to be intersecting.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Crossrefs

These are the Heinz numbers of the partitions counted by A328673.
The strict case is A318719.
The relatively prime version is A328868.
A ranking using binary indices is A326910.
The version for non-isomorphic multiset partitions is A319752.
The version for divisibility (instead of relative primality) is A316476.
Cf. A000837, A056239, A112798, A200976, A289509, A303283, A305843, A318715, A318716, A328336.

Programs

  • Mathematica
    Select[Range[100],And@@(GCD[##]>1&)@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).

Crossrefs

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A318716 Heinz numbers of strict integer partitions with relatively prime parts in which no two parts are relatively prime.

Original entry on oeis.org

2, 17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847, 303949
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sequence of strict integer partitions with Heinz numbers in the sequence begins: (1), (15,10,6), (21,14,6), (20,15,6), (15,12,10), (45,10,6), (18,15,10).

Crossrefs

Cf. A078374, A289509, A302569, A302696, A302796, A302797, A303140, A303280, A303282, A303283, A305713, A318715, A318718, A318719.

Programs

  • Mathematica
    Select[Range[100000],With[{m=PrimePi/@FactorInteger[#][[All,1]]},And[SquareFreeQ[#],GCD@@m==1,And@@(GCD[##]>1&)@@@Select[Tuples[m,2],Less@@#&]]]&]

A318720 Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900
Offset: 1

Views

Author

Gus Wiseman, Sep 02 2018

Keywords

Comments

From Amiram Eldar, Nov 01 2020: (Start)
Also, numbers with more than two non-unitary prime divisors, i.e., numbers k such that A056170(k) > 2, or equivalently, numbers divisible by the squares of three distinct primes.
The complement of the union of A005117, A190641 and A338539.
The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)

Examples

			900 is in the sequence because the factorization 900 = (6*10*15) is relatively prime (since the GCD of (6,10,15) is 1) but each of the pairs (6,10), (6,15), (10,15) has a common divisor > 1. Larger examples are:
1800 = (6*15*20) = (10*12*15).
9900 = (6*10*165) = (6*15*110) = (10*15*66).
5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
60 is not in the sequence because all its possible factorizations (4 * 15, 3 * 4 * 5, etc.) contain at least one pair that is coprime, if not more than one prime.

Links

Crossrefs

Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.
Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.

Programs

  • Mathematica
    strfacs[n_] := If[n <= 1, {{}}, Join@@Table[(Prepend[#1, d] &)/@Select[strfacs[n/d], Min@@#1 > d &], {d, Rest[Divisors[n]]}]]; Select[Range[10000], Function[n, Select[strfacs[n], And[GCD@@# == 1, And@@(GCD[##] > 1 &)@@@Select[Tuples[#, 2], Less@@# &]] &] != {}]]
Select[Range[20000], Count[FactorInteger[#][[;;,2]], ?(#1 > 1 &)] > 2 &] (* _Amiram Eldar, Nov 01 2020 *)

A303365 Number of integer partitions of the n-th squarefree number using squarefree numbers.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 28, 36, 60, 76, 96, 150, 228, 342, 416, 504, 877, 1484, 1759, 2079, 2885, 3387, 3968, 5413, 6304, 7328, 9852, 11395, 13159, 20082, 23056, 39532, 51385, 66488, 85660, 97078, 109907, 140465, 158573, 226918, 255268, 286920, 361606, 405470
Offset: 1

Views

Author

Gus Wiseman, Apr 22 2018

Keywords

Examples

			The a(5) = 9 partitions are (6), (51), (33), (321), (3111), (222), (2211), (21111), (111111).

Links

Crossrefs

Cf. A000051, A005117, A072774, A073576, A078374, A302491, A302796, A303283, A303364.

Programs

  • Mathematica
    nn=80;
sqf=Select[Range[nn],SquareFreeQ];
ser=Product[1/(1-x^sqf[[n]]),{n,Length[sqf]}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,sqf}]

Formula

a(n) = A073576(A005117(n)).
Showing 1-9 of 9 results.

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

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