A294786 -id:A294786 - cp's OEIS frontend

A294788 Number of twice-factorizations of type (Q,P,Q) and product n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 13, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 11, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Gus Wiseman, Nov 08 2017

nonn

a(n) is the number of ways to choose a product-preserving permutation of a set partition of a factorization of n into distinct factors greater than one.

			The a(36) = 13 twice-factorizations are: (2)*(3)*(6), (2)*(3*6), (6)*(2*3), (2*3)*(6), (2*6)*(3), (2*3*6), (2)*(18), (2*18), (3)*(12), (3*12), (4)*(9), (4*9), (36).

Cf. A001055, A063834, A279790, A281113, A294786, A294787.

nn=100;
sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Sum[Times@@Factorial/@Length/@Split[Sort[Times@@@f]],{f,sps[Sort[#]]}]&/@sfs[n]],{n,nn}]

A295279 Number of strict tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 10, 1, 2, 2, 4, 1, 8, 1, 6, 2, 2, 2, 12, 1, 2, 2, 10, 1, 8, 1, 4, 4, 2, 1, 26, 1, 4, 2, 4, 1, 10, 2, 10, 2, 2, 1, 28, 1, 2, 4, 12, 2, 8, 1, 4, 2, 8, 1, 44, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 19 2017

nonn

A strict tree-factorization of n is either (case 1) the number n itself or (case 2) a set of two or more strict tree-factorizations, one of each factor in a factorization of n into distinct factors greater than one.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(30) = 8 strict tree-factorizations are: 30, (2*3*5), (2*15), (2*(3*5)), (3*10), (3*(2*5)), (5*6), (5*(2*3)).
The a(36) = 12 strict tree-factorizations are: 36, (2*3*6), (2*3*(2*3)), (2*18), (2*(2*9)), (2*(3*6)), (2*(3*(2*3))), (3*12), (3*(2*6)), (3*(2*(2*3))), (3*(3*4)), (4*9).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A005804, A045778, A273873, A281113 A281118, A292504, A294786, A295281.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sft[n_]:=1+Total[Function[fac,Times@@sft/@fac]/@Select[sfs[n],Length[#]>1&]];
Array[sft,100]

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w} \\ Andrew Howroyd, Nov 18 2018

a(product of n distinct primes) = A005804(n).
a(prime^n) = A273873(n).
Dirichlet g.f.: (Zeta(s) + Product_{n >= 2}(1 + a(n)/n^s))/2.

A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 29 2017

nonn

By convention a(1) = 1.

Values can be 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 60, 63, 84, 90, etc. - Robert G. Wilson v, Dec 10 2017

			The a(256) = 10 ways are:
(2^1)^8    (2^2)^4   (2^4)^2  (2^8)^1
(4^1)^4    (4^2)^2   (4^4)^1
(16^1)^2   (16^2)^1
(256^1)^1

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

Cf. A000005, A007425, A052409, A052410, A089723, A093771, A175082, A277562, A281113, A284639, A294786, A294336, A294338, A295920, A295923, A295924, A295935.

f:= proc(n) local m,d,t;
  m:= igcd(seq(t[2],t=ifactors(n)[2]));
  add(numtheory:-tau(d),d=numtheory:-divisors(m))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 19 2017

Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

a(A175082(k)) = 1, a(A093771(k)) = 3.

a(n) = Sum_{d|A052409(n)} A000005(d).

A294787 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 12, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 10, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Gus Wiseman, Nov 08 2017

nonn

Cf. A001055, A045778, A279375, A281113, A294617, A294786, A294788.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Total[BellB/@Length/@strfacs[n]],{n,100}]

A294786(n) <= a(n) <= A294788(n).

A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 9, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 12, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 22, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 9, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 12, 1, 5, 1, 7, 5

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(36) = 12 multiset partitions:
  (2*2*3*3)    (6)*(2*3)  (6)*(6)  (36)
  (2*3)*(2*3)  (2*2*9)    (2*18)
               (2*3*6)    (3*12)
               (3*3*4)    (4*9)
                          (6*6)

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A001970, A046523, A050336, A052409, A279375, A294786, A295923, A320886, A320888, A320889.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[facs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A320887(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, binomial(A001055(sqrtnint(n,d)) + d - 1, d))); \\ Antti Karttunen, Nov 17 2019

a(n) = Sum_{d|A052409(n)} binomial(A001055(n^(1/d)) + d - 1, d).

a(n) = a(A046523(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 33, 54, 73, 107, 140, 207, 264, 369, 479, 652, 828, 1112, 1400, 1848, 2326, 3009, 3762, 4856, 6020, 7648, 9478, 11942, 14705, 18427, 22576, 28083, 34350, 42429, 51714, 63680, 77289, 94618, 114648, 139773, 168799, 205144, 247128, 299310, 359958, 434443, 521255, 627812, 751665, 902862

Offset: 0

Gus Wiseman, Oct 23 2018

nonn

			The a(1) = 1 through a(6) = 25 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (12)       (13)          (14)             (15)
       (1)(1)  (111)      (22)          (23)             (24)
               (1)(11)    (112)         (113)            (33)
               (1)(1)(1)  (1111)        (122)            (114)
                          (2)(2)        (1112)           (123)
                          (1)(111)      (11111)          (222)
                          (11)(11)      (2)(12)          (1113)
                          (1)(1)(11)    (1)(1111)        (1122)
                          (1)(1)(1)(1)  (11)(111)        (3)(3)
                                        (1)(1)(111)      (11112)
                                        (1)(11)(11)      (111111)
                                        (1)(1)(1)(11)    (12)(12)
                                        (1)(1)(1)(1)(1)  (2)(112)
                                                         (2)(2)(2)
                                                         (1)(11111)
                                                         (11)(1111)
                                                         (111)(111)
                                                         (1)(1)(1111)
                                                         (1)(11)(111)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(111)
                                                         (1)(1)(11)(11)
                                                         (1)(1)(1)(1)(11)
                                                         (1)(1)(1)(1)(1)(1)

Cf. A001055, A001970, A045778, A050336, A279375, A294617, A294786, A294787, A294788, A320887, A320888, A320889.

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]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Times@@@#&]],{n,8}]

G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M}
a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ Andrew Howroyd, Oct 26 2018

a(13)-a(50) from Andrew Howroyd, Oct 26 2018

A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 8, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 9, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 20 set multipartitions:
  (2*3*4*6)    (2*8*9)     (2*72)     (144)
  (2*6)*(2*6)  (3*6*8)     (3*48)
  (2*6)*(3*4)  (2*3*24)    (4*36)
  (3*4)*(3*4)  (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (12)*(2*6)  (12)*(12)
               (12)*(3*4)

Cf. A001055, A001970, A045778, A050336, A052409, A089259, A294786, A296132, A319269, A320886, A320887, A320889.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[strfacs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]

a(n) = Sum_{d|A052409(n)} binomial(A045778(n^(1/d)) + d - 1, d).

A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 6, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 5, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 17 set partitions:
  (2*3*4*6)    (2*8*9)     (2*72)  (144)
  (2*6)*(3*4)  (3*6*8)     (3*48)
               (2*3*24)    (4*36)
               (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (2*6)*(12)
               (3*4)*(12)

Cf. A000110, A001055, A001970, A045778, A050336, A279375, A294786, A294787, A296132, A320886, A320887, A320888.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Join@@Table[Select[sps[fac],SameQ@@Times@@@#&],{fac,strfacs[n]}]],{n,100}]

A302534 Squarefree numbers whose prime indices are also squarefree and have disjoint prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 41, 43, 47, 51, 55, 58, 59, 62, 66, 67, 73, 79, 82, 83, 85, 86, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 134, 137, 139, 141, 143, 145, 146, 149, 155, 157, 158, 163, 165, 166, 167, 170, 177

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
05: {{2}}
06: {{},{1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
58: {{},{1,3}}
59: {{7}}
62: {{},{5}}
66: {{},{1},{3}}

Cf. A000009, A000961, A001222, A003963, A005117, A007359, A007716, A051424, A056239, A275024, A279375, A281113, A289509, A294786, A301756, A302242, A302243, A302505, A302521.

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

A301756 Number of ways to choose disjoint strict rooted partitions of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 15, 22, 30, 42, 60, 85, 114, 155, 206, 286, 394, 524, 683, 910, 1187, 1564, 2090, 2751, 3543, 4606, 5917, 7598, 9771, 12651, 16260, 20822, 26421, 33525, 42463, 53594, 67337, 85299

Offset: 1

Gus Wiseman, Mar 26 2018

A rooted partition of n is an integer partition of n - 1.

			The a(8) = 10 rooted twice-partitions:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(3)(1)().

Cf. A002865, A032305, A063834, A093637, A275780, A279375, A294786, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],And[UnsameQ@@Total/@#,UnsameQ@@Join@@#]&]//Length,{n,20}]

cp's OEIS Frontend

A294788 Number of twice-factorizations of type (Q,P,Q) and product n.

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A295279 Number of strict tree-factorizations of n.

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A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

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Maple

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A294787 Number of ways to choose a set partition of a factorization of n into distinct factors greater than one.

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A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

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Mathematica

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Extensions

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

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PARI

Extensions

A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

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A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

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