A281116 -id:A281116 - cp's OEIS frontend

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 4, 0, 2, 0, 3, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 0, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 0, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 0, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

The positions of zeros in this sequence are the prime powers A000961.

			The a(144) = 8 aperiodic factorizations are (2*2*2*3*6), (2*2*2*18), (2*2*3*12), (2*3*24), (2*6*12), (2*72), (3*48) and (6*24). Missing from this list are (12*12), (2*2*6*6) and (2*2*2*2*3*3).

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A100953, A275870, A281116, A301700, A303386, A303431, A303546, A303551, A303707, A303709, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]===1&]],{n,100}]

a(n) = Sum_{d in A007916, d|A052409(n)} mu(d) * A303707(n^(1/d)).

A305566 Number of finite sets of relatively prime positive integers > 1 with least common multiple n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 10, 0, 2, 2, 0, 0, 10, 0, 10, 2, 2, 0, 44, 0, 2, 0, 10, 0, 84, 0, 0, 2, 2, 2, 122, 0, 2, 2, 44, 0, 84, 0, 10, 10, 2, 0, 184, 0, 10, 2, 10, 0, 44, 2, 44, 2, 2, 0, 1590, 0, 2, 10, 0, 2, 84, 0, 10, 2, 84, 0, 1156, 0, 2, 10, 10, 2

Offset: 1

Gus Wiseman, Jun 05 2018

nonn

From Robert Israel, Jun 06 2018: (Start)
a(n) depends only on the prime signature of n.
If n is in A000961, a(n)=0.
If n is in A006881, a(n)=2. (End)
If n = p^k*q, where p and q are distinct primes and k >= 1, then a(n) = 3*4^(k-1)-2^(k-1). - Robert Israel, Jun 07 2018

			The a(12) = 10 sets:
{3,4},
{2,3,4}, {2,3,12}, {3,4,6}, {3,4,12},
{2,3,4,6}, {2,3,4,12}, {2,3,6,12}, {3,4,6,12},
{2,3,4,6,12}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000961, A006881, A076078, A181819, A281116, A285572, A290103, A304818, A305563, A305564, A305565, A305567.

f:= proc(n) g(sort(map(t -> t[2],ifactors(n)[2]))) end proc:
f(1):= 0:
g:= proc(L) option remember;
  local nL, Cands, nC, Cons, i;
  nL:= nops(L);
  Cands:= [[]];
  for i from 1 to nL do
    Cands:= [seq(seq([op(s),t],t=0..L[i]),s=Cands)];
  od:
  Cands:= remove(t -> max(t)=0, Cands);
  nC:= nops(Cands);
  Cons:= [seq(select(t -> Cands[t][i]=0, {$1..nC}),i=1..nL),
          seq(select(t -> Cands[t][i]=L[i], {$1..nC}), i=1..nL)];
  h(Cons, {$1..nC})
end proc:
h:= proc(Cons, Cands)
  local t,i,Consi, Candsi;
  if Cons = [] then return 2^nops(Cands) fi;
  t:= 0;
  for i from 1 to nops(Cons[1]) do
    Consi:= map(proc(t) if member(Cons[1][i],t) then NULL else t minus Cons[1][1..i-1] fi end proc, Cons[2..-1]);
    if member({},Consi) then next fi;
    Candsi:= Cands minus Cons[1][1..i];
    t:= t + procname(Consi, Candsi)
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Jun 07 2018

Table[Length[Select[Subsets[Rest[Divisors[n]]],And[GCD@@#==1,LCM@@#==n]&]],{n,100}]

A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 13, 59, 313, 1847, 11977, 84483, 642405, 5228987, 45297249, 415582335, 4021374193, 40895428051, 435721370413, 4850551866619, 56282199807401, 679220819360775, 8508809310177481, 110454586096508563, 1483423600240661781, 20581786429087269819

Offset: 0

Gus Wiseman, Aug 06 2018

nonn

			The a(3) = 13 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,2,2}}, {{1},{2,2}}, {{2},{1,2}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A001055, A007716, A045778, A255906, A281116, A317449, A317532, A317583, A317653, A317752, A317757, A317775.

Row sums of A327116.

C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 16 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@#&]],{n,9}]
(* Second program: *)
c := Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k] c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[b[n, n, i] (-1)^(k-i) c[k, i], {k, 0, n}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

a(0), a(8)-a(22) from Alois P. Heinz, Sep 16 2019

A319055 Maximum product of an integer partition of n with relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 12, 18, 24, 36, 54, 72, 108, 162, 216, 324, 486, 648, 972, 1458, 1944, 2916, 4374, 5832, 8748, 13122, 17496, 26244, 39366, 52488, 78732, 118098, 157464, 236196, 354294, 472392, 708588, 1062882, 1417176, 2125764, 3188646, 4251528, 6377292

Offset: 1

Gus Wiseman, Sep 09 2018

nonn

After a(7), this appears to be the same as A319054.

Cf. A000740, A000792, A000793, A000837, A001414, A007916, A100953, A281116, A289508, A289509, A296302, A319054, A319057.

Table[Max[Times@@@Select[IntegerPartitions[n],GCD@@#==1&]],{n,20}]

A319778 Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 28, 72, 181, 483

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The dual of a multiset partition has empty intersection iff no part contains all the vertices.

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(6) = 13 multiset partitions:
2: {{1},{2}}
3: {{1},{2},{3}}
4: {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,4},{3,4}}
   {{2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}
6: {{3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
   {{1},{2},{1,3},{2,3}}
   {{1},{2},{3,5},{4,5}}
   {{1},{3},{4},{2,3,4}}
   {{1},{3},{2,4},{3,4}}
   {{1},{4},{2,4},{3,4}}
   {{2},{3},{1,3},{2,3}}
   {{2},{4},{1,2},{3,4}}
   {{1},{2},{3},{4},{3,4}}
   {{1},{2},{3},{5},{4,5}}
   {{1},{2},{3},{4},{5},{6}}

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

Cf. A319775, A319779, A319781, A319783.

A293627 Number of knapsack factorizations whose factors sum to n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 19, 21, 27, 34, 45, 51, 69, 77, 100, 117, 146

Offset: 1

Gus Wiseman, Oct 23 2017

A knapsack factorization is a finite multiset of positive integers greater than one such that every distinct submultiset has a different product.

			The a(12) = 19 partitions are:
(12),
(10 2), (9 3), (8 4), (7 5), (6 6),
(8 2 2), (7 3 2), (6 4 2), (6 3 3), (5 5 2), (5 4 3), (4 4 4),
(6 2 2 2), (5 3 2 2), (4 3 3 2), (3 3 3 3),
(3 3 2 2 2),
(2 2 2 2 2 2).

Cf. A000041, A001055, A002033, A002865, A108917, A126796, A275972, A281116, A292886, A294150.

nn=22;
apsQ[y_]:=UnsameQ@@Times@@@Union[Rest@Subsets[y]];
Table[Length@Select[IntegerPartitions[n],apsQ],{n,nn}]

A305150 Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 3, 1, 2, 1, 6, 2, 2, 2, 3, 1, 6, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 5, 1, 3, 5

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 6 factorizations are (3 * 4 * 5), (3 * 20), (4 * 15), (5 * 12), (6 * 10), (60). Missing from this list are (2 * 3 * 10), (2 * 5 * 6), (2 * 30).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A045778, A162247, A259936, A275024, A285572, A281113, A281116, A303386, A303431, A305001, A305148, A305149, A305253.

facs[n_] := If[n <= 1, {{}}, Join@@Table[Map[Prepend[#, d] &, Select[facs[n/d], Min@@ # >= d &]], {d, Rest[Divisors[n]]}]]; Table[Length[Select[facs[n], UnsameQ@@ # && Select[Tuples[Union[#], 2], UnsameQ@@ # && Divisible@@ # &] == {} &]], {n, 100}]

A305150(n, m=n, facs=List([])) = if(1==n, 1, my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&factorback(apply(x -> (x%d),Vec(facs))), newfacs = List(facs); listput(newfacs,d); s += A305150(n/d, d-1, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) <= A045778(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

More terms from Antti Karttunen, Dec 06 2018

A328336 Numbers with no consecutive prime indices relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167

Offset: 1

Gus Wiseman, Oct 14 2019

nonn

First differs from A318978 in having 897, with prime indices {2, 6, 9}.
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), so these are Heinz numbers of partitions no consecutive parts relatively prime (A328187).
Besides the initial 1 this differs from A305078: 47541=897*prime(16) is in A305078 but not in this set. - Andrey Zabolotskiy, Nov 13 2019

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  37: {12}
  39: {2,6}
  41: {13}
  43: {14}

Numbers with consecutive prime indices relatively prime are A328335.
Strict partitions with no consecutive parts relatively prime are A328220.
Numbers with relatively prime prime indices are A289509.
Cf. A000837, A056239, A078374, A112798, A281116, A289508, A318981, A328168, A328169, A328172, A328187, A328188.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MatchQ[primeMS[#],{_,x_,y_,_}/;GCD[x,y]==1]&]

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

A328168 Numbers whose prime indices minus 1 are relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 35, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 95, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 147

Offset: 1

Gus Wiseman, Oct 08 2019

nonn

A multiset is relatively prime if the GCD of its elements is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.
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), so these are Heinz numbers of partitions whose parts minus one are relatively prime. The enumeration of these partitions by sum is given by A328170.

			The sequence of terms together with their prime indices begins:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}
   51: {2,7}
   54: {1,2,2,2}
   57: {2,8}

Positions of 1's in A328167.
Numbers whose prime indices are relatively prime are A289509.
The version for prime indices plus 1 is A318981.
The GCD of prime indices is A289508.
Partitions whose parts minus 1 are relatively prime are A328170.
Cf. A000837, A056239, A112798, A258409, A281116, A290103, A328163, A328169.

q:= n-> igcd(map(i-> numtheory[pi](i[1])-1, ifactors(n)[2])[])=1:
select(q, [$1..150])[];  # Alois P. Heinz, Oct 13 2019

Select[Range[100],GCD@@(PrimePi/@First/@If[#==1,{},FactorInteger[#]]-1)==1&]

cp's OEIS Frontend

A303708 Number of aperiodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

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A305566 Number of finite sets of relatively prime positive integers > 1 with least common multiple n.

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A317776 Number of strict multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers.

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A319055 Maximum product of an integer partition of n with relatively prime parts.

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A319778 Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

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A293627 Number of knapsack factorizations whose factors sum to n.

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A305150 Number of factorizations of n into distinct, pairwise indivisible factors greater than 1.

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PARI

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A328336 Numbers with no consecutive prime indices relatively prime.

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

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