A281113 -id:A281113 - cp's OEIS frontend

A301480 Number of rooted twice-partitions of n.

1, 1, 2, 4, 8, 15, 30, 54, 103, 186, 345, 606, 1115, 1936, 3466, 6046, 10630, 18257, 31927, 54393, 93894, 159631, 272155, 458891, 779375, 1305801, 2196009, 3667242, 6130066, 10170745, 16923127, 27942148, 46211977, 76039205, 125094369, 204952168, 335924597

Offset: 1

Gus Wiseman, Mar 22 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(5) = 8 rooted twice-partitions: ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()).
The a(6) = 15 rooted twice-partitions:
(4), (31), (22), (211), (1111),
(3)(), (21)(), (111)(), (2)(1), (11)(1),
(2)()(), (11)()(), (1)(1)(),
(1)()()(),
()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.

nn=30;
ser=x*Product[1/(1-PartitionsP[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(1/prod(k=1, n-1, 1 - numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} 1/(1 - P(n-1) x^n) where P = A000041.

A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 2, 10, 5, 6, 2, 16, 2, 6, 6, 20, 2, 16, 2, 16, 6, 6, 2, 36, 5, 6, 10, 16, 2, 22, 2, 36, 6, 6, 6, 46, 2, 6, 6, 36, 2, 22, 2, 16, 16, 6, 2, 76, 5, 16, 6, 16, 2, 36, 6, 36, 6, 6, 2, 64, 2, 6, 16, 65, 6, 22, 2, 16, 6, 22, 2, 108, 2, 6, 16, 16, 6

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

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

			The a(6) = 6 factorizations: (2*3)*(), (3)*(2), (2)*(3), ()*(2*3), (6)*(), ()*(6).
The a(12) = 16 factorizations:
  ()*(2*2*3), (2)*(2*3), (3)*(2*2), (2*2)*(3), (2*3)*(2), (2*2*3)*(),
  ()*(2*6), (2)*(6), (6)*(2), (2*6)*(), ()*(3*4), (3)*(4), (4)*(3), (3*4)*(),
  ()*(12), (12)*().

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Index to sequences related to prime signature

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301829.

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

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={MultEulerT(vector(n, i, 2))} \\ Andrew Howroyd, Nov 18 2018

Dirichlet g.f.: Product_{n > 1} 1/(1 - n^(-s))^2. [corrected by Ilya Gutkovskiy, Dec 14 2020]

a(p^n) = A000712(n) for prime p. - Andrew Howroyd, Nov 18 2018

A302492 Products of any power of 2 with prime numbers of prime-power index, i.e., prime numbers p of the form p = prime(q^k), for q prime, k >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 72, 75, 76, 77, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Apr 08 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 multiset multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A007716, A050361, A056239, A076610, A270995, A275024, A279784, A281113, A295935, A301762, A302242, A302243, A302493.

Select[Range[100],Or[#===1,And@@PrimePowerQ/@PrimePi/@DeleteCases[FactorInteger[#][[All,1]],2]]&]

ok(n)={!#select(p->p<>2&&!isprimepower(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A300486 Number of relatively prime or monic partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 15, 18, 28, 35, 56, 64, 101, 120, 168, 210, 297, 348, 490, 583, 776, 946, 1255, 1482, 1952, 2335, 2981, 3581, 4565, 5387, 6842, 8119, 10086, 12013, 14863, 17527, 21637, 25525, 31083, 36695, 44583, 52256, 63261, 74171, 88932, 104303, 124754

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

			The a(6) = 8 relatively prime or monic partitions are (6), (51), (411), (321), (3111), (2211), (21111), (111111). Missing from this list are (42), (33), (222).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A301462, A301467, A301480, A302094, A302698, A302915, A302916, A302917.

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

a(n)={(n > 1) + sumdiv(n, d, moebius(d)*numbpart(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n > 1) = 1 + A000837(n) = 1 + Sum_{d|n} mu(d) * A000041(n/d).

A281119 Number of complete tree-factorizations of n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 9, 1, 1, 2, 3, 1, 4, 1, 12, 1, 1, 1, 12, 1, 1, 1, 9, 1, 4, 1, 3, 3, 1, 1, 29, 1, 3, 1, 3, 1, 9, 1, 9, 1, 1, 1, 17, 1, 1, 3, 34, 1, 4, 1, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1

Offset: 2

Gus Wiseman, Jan 15 2017

nonn

A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors greater than 1. A complete (or total) tree-factorization is a tree-factorization whose leaves are all prime numbers.

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

			The a(36)=12 complete tree-factorizations of 36 are:
(2(2(33))), (2(3(23))), (2(233)),   (3(2(23))),
(3(3(22))), (3(223)),   ((22)(33)), ((23)(23)),
(22(33)),   (23(23)),   (33(22)),   (2233).

Michael De Vlieger, Table of n, a(n) for n = 2..10000
A. Knopfmacher, M. Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal 10(1), 2006.

Cf. A000311, A001055, A063834, A196545, A262673, A273873, A281113, A281118.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1,{{}},Prepend[Join@@Function[q,Tuples[treefacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[Select[treefacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,83}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=w[k]^e*v[i]))); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A196545(n) for prime p. - Andrew Howroyd, Nov 18 2018

A301706 Number of rooted thrice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983

Offset: 1

Gus Wiseman, Mar 25 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

			The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
Array[thrire,30]

A066815 Number of partitions of n into sums of products.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 33, 45, 69, 94, 148, 197, 289, 390, 575, 762, 1086, 1439, 2040, 2687, 3712, 4874, 6749, 8792, 11918, 15526, 20998, 27164, 36277, 46820, 62367, 80146, 105569, 135326, 177979, 227139, 296027, 377142, 490554, 622526, 804158

Offset: 0

Vladeta Jovovic, Jan 20 2002

nonn

Number of ways to choose a factorization of each part of an integer partition of n. - Gus Wiseman, Sep 05 2018

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = A001055(n). - Seiichi Manyama, Nov 14 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 14 partitions of 6 into sums of products:
  6, 2*3,
  5+1, 4+2, 2*2+2, 3+3,
  4+1+1, 2*2+1+1, 3+2+1, 2+2+2,
  3+1+1+1, 2+2+1+1,
  2+1+1+1+1,
  1+1+1+1+1+1.
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000041, A001055, A066739, A321460.

Cf. A001970, A050336, A063834, A065026, A281113, A284639, A318948, A318949.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]],{n,20}] (* Gus Wiseman, Sep 05 2018 *)

G.f.: Product_{k>=1} 1/(1-A001055(k)*x^k).

a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n > 0, a(0)=1, b(k)=Sum_{d|k} d*(A001055(d))^(k/d).

Renamed by T. D. Noe, May 24 2011

A302491 Prime numbers of squarefree index.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 101, 109, 113, 127, 137, 139, 149, 157, 163, 167, 179, 181, 191, 199, 211, 233, 241, 257, 269, 271, 277, 283, 293, 313, 317, 331, 347, 349, 353, 367, 373, 389, 397, 401, 421, 431, 439, 443, 449, 461

Offset: 1

Gus Wiseman, Apr 08 2018

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

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

Cf. A000040, A000961, A001222, A005117, A007097, A056239, A063834, A275024, A281113, A302242, A302478.

map(ithprime, select(numtheory:-issqrfree, [$1..500])); # Robert Israel, Nov 06 2023

Mathematica
```
Prime/@Select[Range[100],SquareFreeQ]
```

forprime(p=1, 500, if(issquarefree(primepi(p)), print1(p, ", "))) \\ Felix Fröhlich, Apr 10 2018

list(lim)=my(v=List(),k); forprime(p=2,lim\1, if(issquarefree(k++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 03 2023

a(n) = A000040(A005117(n)).

a(n) ~ kn log n, where k = Pi^2/6. - Charles R Greathouse IV, Aug 03 2023

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.

facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[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,_}];
mps[set_]:=Union[Sort[Sort /@ (#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A300335 Number of ordered set partitions of {1,...,n} with weakly increasing block-sums.

Original entry on oeis.org

1, 1, 2, 6, 18, 65, 258, 1156, 5558, 29029, 161942, 967921, 6110687, 40807420, 286177944, 2107745450, 16202590638, 130043111849, 1085011337141, 9408577992091, 84501248359552, 786018565954838, 7550153439748394

Offset: 0

Gus Wiseman, Mar 03 2018

			The a(3) = 6 ordered set partitions: (123), (1)(23), (2)(13), (12)(3), (3)(12), (1)(2)(3).

A000110(n) <= a(n) <= A000670(n).

Cf. A005651, A063834, A275780, A279375, A279790, A279791, A281113.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@GatherBy[sptn,Total],{sptn,sps[Range[n]]}],{n,8}]

a(12)-a(15) from Alois P. Heinz, Mar 03 2018

a(16)-a(22) from Christian Sievers, Aug 30 2024

cp's OEIS Frontend

A301480 Number of rooted twice-partitions of n.

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A301830 Number of factorizations of n into factors (greater than 1) of two kinds.

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A302492 Products of any power of 2 with prime numbers of prime-power index, i.e., prime numbers p of the form p = prime(q^k), for q prime, k >= 1.

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A300486 Number of relatively prime or monic partitions of n.

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A281119 Number of complete tree-factorizations of n >= 2.

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A301706 Number of rooted thrice-partitions of n.

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A066815 Number of partitions of n into sums of products.

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