A318715 -id:A318715 - cp's OEIS frontend

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

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

Gus Wiseman, Sep 02 2018

nonn

			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).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600

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

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

a(51)-a(69) from Alois P. Heinz, Sep 02 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@@#&]]&]

A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 13, 49, 199

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(8) = 13 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}

Cf. A007716, A283877, A305854, A306006, A317757, A318715, A318717.

Cf. A319752, A319762, A319763, A319764, A319765, A319779, A319781.

A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 3, 9, 21, 48, 103, 214, 436, 863, 1689

Offset: 0

Gus Wiseman, Sep 27 2018

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

			The a(3) = 1 through a(5) = 9 multiset partitions:
3: {{1},{2}}
4: {{1},{3}}
   {{2},{1,1}}
   {{1},{1},{2}}
5: {{1},{4}}
   {{2},{3}}
   {{3},{1,1}}
   {{1},{2,2}}
   {{1},{1},{3}}
   {{1},{2},{2}}
   {{2},{1,1,1}}
   {{1},{2},{1,1}}
   {{1},{1},{1},{2}}

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

Cf. A319775, A319779, A319778, A319783.

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}]&]

A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

Original entry on oeis.org

0, 52, 116, 772, 832, 836, 1072, 1076, 1136, 1140, 1796, 1856, 1860, 2320, 2368, 2384, 2592, 2624, 2656, 2880, 3088, 3104, 3120, 3136, 3152, 3168, 3184, 3344, 3392, 3408, 3616, 3648, 3680, 3904, 4132, 4148, 4196, 4212, 4612, 4640, 4644, 4672, 4676, 4704, 4708

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all pairwise intersecting set-systems with empty intersection, together with their BII-numbers, begins:
     0: {}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   772: {{1,2},{1,4},{2,4}}
   832: {{1,2,3},{1,4},{2,4}}
   836: {{1,2},{1,2,3},{1,4},{2,4}}
  1072: {{1,3},{2,3},{1,2,4}}
  1076: {{1,2},{1,3},{2,3},{1,2,4}}
  1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
  1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
  1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
  2320: {{1,3},{1,4},{3,4}}
  2368: {{1,2,3},{1,4},{3,4}}
  2384: {{1,3},{1,2,3},{1,4},{3,4}}
  2592: {{2,3},{2,4},{3,4}}
  2624: {{1,2,3},{2,4},{3,4}}
  2656: {{2,3},{1,2,3},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}

Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326911.

Cf. A318715, A319759, A319762, A319763, A319764.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],(#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]

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

Gus Wiseman, Sep 02 2018

nonn

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

			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).

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

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

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

Cf. A319748, A319755, A319778, A319781, A319790.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t] - subst(x^(t*k)*u[t] + O(x*x^(n\2)), x, x^2), O(x*x^n) ))*if(k,1+x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A202425 Number of partitions of n into parts having pairwise common factors but no overall common factor.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 1, 6, 0, 5, 0, 2, 2, 9, 0, 8, 2, 4, 3, 16, 0, 22, 5, 6, 5, 19, 2, 35, 8, 14, 6, 44, 4, 55, 13, 16, 19, 64, 6, 82, 17, 39, 31, 108, 10, 105, 40, 66, 46, 161, 14, 182, 61, 97, 72, 207, 37, 287, 85, 144, 93, 357, 59

Offset: 31

Alois P. Heinz, Dec 19 2011

nonn

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(37) = 2: [6,6,10,15], [10,12,15].
a(41) = 3: [6,10,10,15], [6,15,20], [6,14,21].
a(47) = 6: [6,6,10,10,15], [10,10,12,15], [6,6,15,20], [12,15,20], [6,6,14,21], [12,14,21].
a(49) = 5: [6,6,6,6,10,15], [6,6,10,12,15], [10,12,12,15], [6,10,15,18], [10,15,24].

Fausto A. C. Cariboni, Table of n, a(n) for n = 31..400 (terms 31..251 from Alois P. Heinz)

Cf. A202385, A018783.
The version with only distinct parts compared is A328672.
The Heinz numbers of these partitions are A328868.
The strict case is A202385, which is essentially the same as A318715.
The version for non-isomorphic multiset partitions is A319759.
The version for set-systems is A326364.
Intersecting partitions are A200976.
Cf. A000837, A305148, A305843, A306006, A318717, A328673, A328867.

with(numtheory):
w:= (m, h)-> mul(`if`(j>=h, 1, j), j=factorset(m)):
b:= proc(n, i, g, s) option remember; local j, ok, si;
      if n<0 then 0
    elif n=0 then `if`(g>1, 0, 1)
    elif i<2 or member(1, s) then 0
    else ok:= evalb(i<=n);
         si:= map(x->w(x, i), s);
         for j in s while ok do ok:= igcd(i, j)>1 od;
         b(n, i-1, g, si) +`if`(ok, add(b(n-t*i, i-1, igcd(i, g),
                      si union {w(i,i)} ), t=1..iquo(n, i)), 0)
      fi
    end:
a:= n-> b(n, n, 0, {}):
seq(a(n), n=31..100);

w[m_, h_] := Product[If[j >= h, 1, j], {j, FactorInteger[m][[All, 1]]}]; b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n<0, 0, n == 0, If[g>1, 0, 1], i<2 || MemberQ[s, 1], 0, True, ok = (i <= n); si = w[#, i]& /@ s; Do[If[ok, ok = (GCD[i, j]>1)], {j, s}]; b[n, i-1, g, si] + If[ok, Sum[b[n-t*i, i-1, GCD[i, g], si ~Union~ {w[i, i]}], {t, 1, Quotient[n, i]}], 0]]]; a[n_] := b[n, n, 0, {}]; Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And@@(GCD[##]>1&)@@@Tuples[#,2]&]],{n,0,40}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328672(n) - 1. - Gus Wiseman, Nov 04 2019

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

Gus Wiseman, Sep 02 2018

nonn

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)

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

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 *)

cp's OEIS Frontend

A318717 Number of strict integer partitions of n 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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A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

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A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

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

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Mathematica

A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

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Mathematica

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

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Mathematica

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

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PARI

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A202425 Number of partitions of n into parts having pairwise common factors but no overall common factor.

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Maple