A318716 -id:A318716 - cp's OEIS frontend

A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 5, 0, 0, 0, 1, 0, 5, 0, 1, 0, 6, 0, 6, 0, 0, 0, 4, 0, 6, 0, 0, 0, 9, 0, 2, 1, 2, 0, 9, 0, 8, 1, 1, 0, 5, 0, 14, 0, 1, 0, 15, 0, 14, 0, 0, 1, 14, 0, 14, 0, 2, 0, 15, 0, 6, 1, 2, 1, 11, 0, 18, 1, 1, 0, 10, 0, 23

Offset: 31

Alois P. Heinz, Apr 03 2017

The Heinz numbers of these partitions are the intersection of A014612 (triples), A289509 (relatively prime), and A337694 (pairwise non-coprime). - Gus Wiseman, Oct 16 2020

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(41) = 2: [6,14,21], [6,15,20].
From _Gus Wiseman_, Oct 14 2020: (Start)
Selected terms and the corresponding triples:
  a(31)=1: a(41)=2: a(59)=3:  a(77)=4:  a(61)=5:  a(71)=6:
-------------------------------------------------------------
  15,10,6  20,15,6  24,20,15  39,26,12  33,22,6   39,26,6
           21,14,6  24,21,14  42,20,15  40,15,6   45,20,6
                    35,14,10  45,20,12  45,10,6   50,15,6
                              50,15,12  28,21,12  35,21,15
                                        36,15,10  36,20,15
                                                  36,21,14
(End)

Alois P. Heinz, Table of n, a(n) for n = 31..10000

Cf. A082024, A230034, A230035.
A023023 does not require pairwise non-coprimality, with strict case A101271.
A202425 and A328672 count these partitions of any length, ranked by A328868.
A284825*6 is the ordered version.
A307719 is the pairwise coprime instead of non-coprime version.
A337599 does not require relatively primality, with strict case A337605.
A200976 and A328673 count pairwise non-coprime partitions.
A289509 gives Heinz numbers of relatively prime partitions.
A327516 counts pairwise coprime partitions, ranked by A333227.
A337694 gives Heinz numbers of pairwise non-coprime partitions.
Cf. A000217, A000741, A001399, A007304, A014612, A220377, A318716, A328679, A337604, A337666.

a:= proc(n) option remember; add(add(`if`(igcd(i, j)>1
      and igcd(i, j, n-i-j)=1 and igcd(i, n-i-j)>1 and
      igcd(j, n-i-j)>1, 1, 0), j=i..(n-i)/2), i=2..n/3)
    end:
seq(a(n), n=31..137);

a[n_] := a[n] = Sum[Sum[If[GCD[i, j] > 1 && GCD[i, j, n - i - j] == 1 && GCD[i, n - i - j] > 1 && GCD[j, n - i - j] > 1, 1, 0], {j, i, (n - i)/2} ], {i, 2, n/3}];
Table[a[n], {n, 31, 137}] (* Jean-François Alcover, Jun 13 2018, from Maple *)
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&&stabQ[#,CoprimeQ]&]],{n,31,100}] (* Gus Wiseman, Oct 14 2020 *)

a(n) > 0 iff n in { A230035 }.

a(n) = 0 iff n in { A230034 }.

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

Gus Wiseman, Sep 02 2018

nonn

			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).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500

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

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

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

Gus Wiseman, Sep 02 2018

nonn

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

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

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

Gus Wiseman, Oct 30 2019

nonn

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.

			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}

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.

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

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Oct 31 2020

nonn

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 partitions whose set of distinct parts is a singleton or pairwise coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
      2: {1}         17: {7}           33: {2,5}
      3: {2}         18: {1,2,2}       34: {1,7}
      4: {1,1}       19: {8}           35: {3,4}
      5: {3}         20: {1,1,3}       36: {1,1,2,2}
      6: {1,2}       22: {1,5}         37: {12}
      7: {4}         23: {9}           38: {1,8}
      8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
      9: {2,2}       25: {3,3}         41: {13}
     10: {1,3}       26: {1,6}         43: {14}
     11: {5}         27: {2,2,2}       44: {1,1,5}
     12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
     13: {6}         29: {10}          46: {1,9}
     14: {1,4}       30: {1,2,3}       47: {15}
     15: {2,3}       31: {11}          48: {1,1,1,1,2}

A302798 is the squarefree case.
A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
A304711 does not consider singletons relatively prime, except for (1).
A304712 counts the partitions with these Heinz numbers.
A316476 is the version for indivisibility instead of relative primality.
A328867 is the pairwise non-coprime instead of pairwise coprime version.
A337600 counts triples of this type, with ordered version A337602.
A338330 is the complement.
A000961 lists powers of primes.
A051424 counts pairwise coprime or singleton partitions.
A304038 gives the distinct prime indices of each positive integer.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.

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

Equals A304711 \/ A000961.

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

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

			The sequence of terms together with their prime indices begins:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

A319319 Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 167, 173, 177

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

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

First differs from A304713 (Heinz numbers of pairwise indivisible partitions) at A304713(58) = 165, which is absent from this sequence because its prime indices are {2,3,5} and GCD(2,3) = GCD(2,3,5) = 1. The first term with more than two prime factors is 17719, which has prime indices {6,10,15}. The first term with more than two prime factors that is absent from A318716 is 296851, which has prime indices {12,20,30}.

			The sequence of partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (3), (4), (5), (6), (3,2), (7), (8), (9), (10), (11), (5,2), (4,3), (12), (13), (14), (15), (7,2), (16), (5,3).

Cf. A056239, A108917, A122768, A275972, A299702, A301899, A301900, A304713, A316313, A319315, A319318, A319327.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@GCD@@@Union[Subsets[primeMS[#]]]&]

A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 17719, 32768, 40807, 43381, 50431, 65536, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 131072, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

Equals the union A000079 and A328868.
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.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     16: {1,1,1,1}
     32: {1,1,1,1,1}
     64: {1,1,1,1,1,1}
    128: {1,1,1,1,1,1,1}
    256: {1,1,1,1,1,1,1,1}
    512: {1,1,1,1,1,1,1,1,1}
   1024: {1,1,1,1,1,1,1,1,1,1}
   2048: {1,1,1,1,1,1,1,1,1,1,1}
   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
  16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  17719: {6,10,15}
  32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  40807: {6,14,21}
  43381: {6,15,20}
  50431: {10,12,15}
  65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328672.
Terms that are not powers of 2 are A328868.
The strict case is A318716.
The version without global relative primality is A328867.
A ranking using binary indices (instead of prime indices) is A326912.
The version for non-isomorphic multiset partitions is A319759.
The version for divisibility (instead of relative primality) is A328677.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A202425, A289509, A291166, A302796, A316476, A318715, A319752, A328336.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],#==1||GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[primeMS[#]],{2}]]&]

A338330 Numbers that are neither a power of a prime (A000961) nor is their set of distinct prime indices pairwise coprime.

Original entry on oeis.org

21, 39, 42, 57, 63, 65, 78, 84, 87, 91, 105, 111, 114, 115, 117, 126, 129, 130, 133, 147, 156, 159, 168, 171, 174, 182, 183, 185, 189, 195, 203, 210, 213, 222, 228, 230, 231, 234, 235, 237, 247, 252, 258, 259, 260, 261, 266, 267, 273, 285, 294, 299, 301

Offset: 1

Gus Wiseman, Nov 12 2020

nonn

Also Heinz numbers of partitions that are neither constant (A144300) nor have pairwise coprime distinct parts (A304709), hence the formula. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     21: {2,4}        126: {1,2,2,4}      203: {4,10}
     39: {2,6}        129: {2,14}         210: {1,2,3,4}
     42: {1,2,4}      130: {1,3,6}        213: {2,20}
     57: {2,8}        133: {4,8}          222: {1,2,12}
     63: {2,2,4}      147: {2,4,4}        228: {1,1,2,8}
     65: {3,6}        156: {1,1,2,6}      230: {1,3,9}
     78: {1,2,6}      159: {2,16}         231: {2,4,5}
     84: {1,1,2,4}    168: {1,1,1,2,4}    234: {1,2,2,6}
     87: {2,10}       171: {2,2,8}        235: {3,15}
     91: {4,6}        174: {1,2,10}       237: {2,22}
    105: {2,3,4}      182: {1,4,6}        247: {6,8}
    111: {2,12}       183: {2,18}         252: {1,1,2,2,4}
    114: {1,2,8}      185: {3,12}         258: {1,2,14}
    115: {3,9}        189: {2,2,2,4}      259: {4,12}
    117: {2,2,6}      195: {2,3,6}        260: {1,1,3,6}

A338331 is the complement.
A304713 is the complement of the version for divisibility.
Cf. A056239, A112798, A289509, A303282, A316476, A318716, A318719, A328867.

Select[Range[2,100],!PrimePowerQ[#]&&!CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

Equals A024619 \ A304711.

A338552 Non-powers of primes whose prime indices have a common divisor > 1.

Original entry on oeis.org

21, 39, 57, 63, 65, 87, 91, 111, 115, 117, 129, 133, 147, 159, 171, 183, 185, 189, 203, 213, 235, 237, 247, 259, 261, 267, 273, 299, 301, 303, 305, 319, 321, 325, 333, 339, 351, 365, 371, 377, 387, 393, 399, 417, 427, 441, 445, 453, 477, 481, 489, 497, 507

Offset: 1

Gus Wiseman, Nov 03 2020

nonn

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 non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     21: {2,4}      183: {2,18}       305: {3,18}
     39: {2,6}      185: {3,12}       319: {5,10}
     57: {2,8}      189: {2,2,2,4}    321: {2,28}
     63: {2,2,4}    203: {4,10}       325: {3,3,6}
     65: {3,6}      213: {2,20}       333: {2,2,12}
     87: {2,10}     235: {3,15}       339: {2,30}
     91: {4,6}      237: {2,22}       351: {2,2,2,6}
    111: {2,12}     247: {6,8}        365: {3,21}
    115: {3,9}      259: {4,12}       371: {4,16}
    117: {2,2,6}    261: {2,2,10}     377: {6,10}
    129: {2,14}     267: {2,24}       387: {2,2,14}
    133: {4,8}      273: {2,4,6}      393: {2,32}
    147: {2,4,4}    299: {6,9}        399: {2,4,8}
    159: {2,16}     301: {4,14}       417: {2,34}
    171: {2,2,8}    303: {2,26}       427: {4,18}

A318978 allows prime powers, counted by A018783, with complement A289509.
A327685 allows nonprime prime powers.
A338330 is the coprime instead of relatively prime version.
A338554 counts the partitions with these Heinz numbers.
A338555 is the complement.
A000740 counts relatively prime compositions.
A000961 lists powers of primes, with complement A024619.
A051424 counts pairwise coprime or singleton partitions.
A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
A291166 ranks relatively prime compositions, with complement A291165.
A302696 gives the Heinz numbers of pairwise coprime partitions.
A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.

Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]

Equals A024619 /\ A318978.

Complement of A000961 \/ A289509.

cp's OEIS Frontend

A284825 Number of partitions of n into 3 parts without common divisors such that every pair of them has common divisors.

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

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

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

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A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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A319319 Heinz numbers of integer partitions such that every distinct submultiset has a different GCD.

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A328679 Heinz numbers of integer partitions with no two distinct parts relatively prime, but with no divisor in common to all of the parts.

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A338330 Numbers that are neither a power of a prime (A000961) nor is their set of distinct prime indices pairwise coprime.

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