A281118 -id:A281118 - cp's OEIS frontend

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(6) = 6 tree-partitions of {1,1,2}:
  (112)
  ((1)(12))
  ((2)(11))
  ((1)(1)(2))
  ((1)((1)(2)))
  ((2)((1)(1)))

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[allmsptrees[nrmptn[n]]],{n,20}]

a(n) = A281118(A181821(n)).
a(prime(n)) = A289501(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

A319118 Number of multimin tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 8, 1, 2, 2, 24, 1, 6, 1, 8, 2, 2, 1, 42, 2, 2, 6, 8, 1, 8, 1, 112, 2, 2, 2, 38, 1, 2, 2, 42, 1, 8, 1, 8, 8, 2, 1, 244, 2, 6, 2, 8, 1, 24, 2, 42, 2, 2, 1, 58, 1, 2, 8, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24

Offset: 1

Gus Wiseman, Sep 10 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.

			The a(12) = 8 multimin tree-factorizations:
  12,
  (2*6), (4*3), (6*2), (2*2*3),
  (2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
  112,
  (1,12), (11,2), (12,1), (1,1,2),
  (1,(1,2)), ((1,1),2), ((1,2),1).

Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[mmftrees[n]],{n,100}]

a(prime^n) = A118376(n).

a(product of n distinct primes) = A005804(n).

A319137 Number of strict planar branching factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 3, 7, 1, 9, 1, 9, 3, 3, 1, 37, 1, 3, 3, 9, 1, 25, 1, 21, 3, 3, 3, 57, 1, 3, 3, 37, 1, 25, 1, 9, 9, 3, 1, 161, 1, 9, 3, 9, 1, 37, 3, 37, 3, 3, 1, 153, 1, 3, 9, 75, 3, 25, 1, 9, 3, 25, 1, 345, 1, 3, 9, 9, 3, 25, 1, 161

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n.

			The a(12) = 9 trees:
  12,
  (2*6), (3*4), (4*3),(6*2),
  (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).

Cf. A000108, A001055, A045778, A074206, A118376, A277130, A281113, A281118, A292504, A295279, A295281, A317144, A319122, A319123, A319138.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[sotfs[n]],{n,100}]

a(prime^n) = A319123(n + 1).

a(product of n distinct primes) = A319122(n).

A319138 Number of complete strict planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 4, 1, 2, 2, 0, 1, 4, 1, 4, 2, 2, 1, 8, 0, 2, 0, 4, 1, 18, 1, 0, 2, 2, 2, 28, 1, 2, 2, 8, 1, 18, 1, 4, 4, 2, 1, 16, 0, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 84, 1, 2, 4, 0, 2, 18, 1, 4, 2, 18, 1, 112, 1, 2, 4, 4, 2, 18, 1, 16, 0, 2, 1

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n. A strict planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees: (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).

Cf. A000108, A045778, A074206, A118376, A277130, A281113, A281118, A295279, A295281, A317144, A319122, A319123, A319136, A319137.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[Select[sotfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A000007(n - 1).

a(product of n distinct primes) = A032037(n).

A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319240.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319237.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],_Integer?(Abs[#]>0&)]

A316789 Number of same-tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 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, Jul 14 2018

nonn

A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). A same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite sequence of two or more same-tree-factorizations, one of each factor in a constant factorization of n.

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

			The a(64) = 14 same-tree-factorizations:
  64
  (8*8)
  (4*4*4)
  (8*(2*2*2))
  ((2*2*2)*8)
  (4*4*(2*2))
  (4*(2*2)*4)
  ((2*2)*4*4)
  (2*2*2*2*2*2)
  (4*(2*2)*(2*2))
  ((2*2)*4*(2*2))
  ((2*2)*(2*2)*4)
  ((2*2*2)*(2*2*2))
  ((2*2)*(2*2)*(2*2))

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

Cf. A001055, A001597, A001678, A003238, A007916, A052409, A052410, A067824, A089723, A281118, A281145, A294336, A316790.

a[n_]:=1+Sum[a[n^(1/d)]^d,{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, a(z^(e/d))^d)))} \\ Andrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = x^y, y > 1} a(x)^y.

a(2^n) = A281145(n).

A316790 Number of orderless same-tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 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, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 14 2018

nonn

A constant factorization of n is a finite nonempty constant multiset of positive integers greater than 1 with product n. Constant factorizations correspond to perfect divisors (A089723). An orderless same-tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless same-tree-factorizations, one of each factor in a constant factorization of n.

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

			The a(64) = 9 orderless same-tree-factorizations:
  64
  (8*8)
  (4*4*4)
  (4*4*(2*2))
  (8*(2*2*2))
  (2*2*2*2*2*2)
  (4*(2*2)*(2*2))
  ((2*2*2)*(2*2*2))
  ((2*2)*(2*2)*(2*2))

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

Cf. A001055, A001597, A003238, A052409, A052410, A067824, A089723, A281118, A281145, A289078, A292504, A316789.

a[n_]:=1+Sum[Binomial[a[n^(1/d)]+d-1,d],{d,Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]]}]
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(d>1, binomial(a(z^(e/d)) + d - 1, d))))} \\ Andrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = x^y, y > 1} binomial(a(x) + y - 1, y).

a(2^n) = A289078(n).

A318577 Number of complete multimin tree-factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 11, 1, 3, 1, 4, 1, 1, 1, 19, 1, 1, 3, 4, 1, 4, 1, 45, 1, 1, 1, 17, 1, 1, 1, 19, 1, 4, 1, 4, 4, 1, 1, 96, 1, 3, 1, 4, 1, 11, 1, 19, 1, 1, 1, 26, 1, 1, 4, 197, 1, 4, 1, 4, 1, 4, 1, 104, 1, 1, 3, 4, 1, 4, 1, 96, 11, 1, 1, 26, 1, 1, 1, 19, 1, 19, 1, 4, 1, 1, 1, 501, 1, 3, 4, 17

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees are (2*2*3), (2*(2*3)), ((2*3)*2), ((2*2)*3).

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A319118, A319121.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[Select[mmftrees[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A000311(n).

A319119 Number of multimin tree-factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 3, 9, 37, 173, 921, 5185, 30497, 184469, 1140413, 7170085, 45704821

Offset: 1

Gus Wiseman, Sep 10 2018

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.

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

			The a(3) = 9 multimin tree-factorizations:
  5, 6, 8,
  (2*3), (2*4), (4*2), (2*2*2),
  (2*(2*2)), ((2*2)*2).
Or as series-reduced plane trees of multisets:
  3, 12, 111,
  (1,2), (1,11), (11,1), (1,1,1),
  (1,(1,1)), ((1,1),1).

Cf. A001055, A020639, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A319118.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Sum[Length[mmftrees[k]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,7}]

a(11)-a(12) from Robert Price, Sep 14 2018

A319136 Number of complete planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 2, 11, 1, 9, 1, 9, 2, 2, 1, 44, 1, 2, 3, 9, 1, 18, 1, 45, 2, 2, 2, 66, 1, 2, 2, 44, 1, 18, 1, 9, 9, 2, 1, 225, 1, 9, 2, 9, 1, 44, 2, 44, 2, 2, 1, 132, 1, 2, 9, 197, 2, 18, 1, 9, 2, 18, 1, 450, 1, 2, 9, 9, 2, 18, 1, 225

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 9 trees:
  (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).

Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A032037(n).

cp's OEIS Frontend

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

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A319118 Number of multimin tree-factorizations of n.

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A319137 Number of strict planar branching factorizations of n.

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A319138 Number of complete strict planar branching factorizations of n.

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A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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A316789 Number of same-tree-factorizations of n.

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A316790 Number of orderless same-tree-factorizations of n.

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A318577 Number of complete multimin tree-factorizations of n.

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