A162247 -id:A162247 - cp's OEIS frontend

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

Offset: 1

Gus Wiseman, May 11 2021

Differs from A339195 in having 77 before 66.

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A372053 Irregular array read by rows: row n lists the factorizations of n into a product of nondecreasing integers >= 2.

Original entry on oeis.org

2, 3, 2, 2, 4, 5, 2, 3, 6, 7, 2, 2, 2, 2, 4, 8, 3, 3, 9, 2, 5, 10, 11, 2, 2, 3, 2, 6, 3, 4, 12, 13, 2, 7, 14, 3, 5, 15, 2, 2, 2, 2, 2, 2, 4, 2, 8, 4, 4, 16, 17, 2, 3, 3, 2, 9, 3, 6, 18, 19, 2, 2, 5, 2, 10, 4, 5, 20, 3, 7, 21, 2, 11, 22, 23, 2, 2, 2, 3, 2, 2, 6, 2, 3, 4, 2, 12, 3, 8, 4, 6, 24, 5, 5, 25, 2, 13, 26, 3, 3, 3, 3, 9, 27, 2, 2, 7, 2, 14, 4, 7, 28, 29, 2, 3, 5, 2, 15, 3, 10, 5, 6, 30

Offset: 2

N. J. A. Sloane, Apr 22 2024, following a suggestion from Scott R. Shannon

Factorizations of n are ordered lexicographically. Compare A162247.

			The factorizations of the numbers 2 through 24 are:
  2;
  3;
  2, 2;  4;
  5;
  2, 3;  6;
  7;
  2, 2, 2;  2, 4;  8;
  3, 3;  9;
  2, 5;  10;
  11;
  2, 2, 3;  2, 6;  3, 4;  12;
  13;
  2, 7;  14;
  3, 5;  15;
  2, 2, 2, 2;  2, 2, 4;  2, 8;  4, 4;  16;
  17;
  2, 3, 3;  2, 9;  3, 6;  18;
  19;
  2, 2, 5;  2, 10;  4, 5;  20;
  3, 7;  21;
  2, 11;  22;
  23;
  2, 2, 2, 3;  2, 2, 6;  2, 3, 4;  2, 12;  3, 8;  4, 6;  24;

Michael De Vlieger, Table of n, a(n) for n = 2..19795 (rows n = 1..1000, flattened)

Cf. A001055, A027746, A066637 (row lengths), A162247.

f[x_] := If[x <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[x/d], Min @@ # >= d &]], {d, Rest@ Divisors[x]}]]; Array[Flatten @* f, 29, 2] // Flatten (* Michael De Vlieger, Apr 22 2024 *)

The DATA section is longer than usual in order to show the factorizations of 30.

Edited by Peter Munn, Feb 26 2025

A122819 Array read by rows in which the n-th row contains smallest odd numbers in increasing order of all possible prime signatures with n divisors.

Original entry on oeis.org

1, 3, 9, 15, 27, 81, 45, 243, 729, 105, 135, 2187, 225, 6561, 405, 19683, 59049, 315, 675, 1215, 177147, 531441, 3645, 1594323, 2025, 4782969, 945, 1155, 3375, 10935, 14348907, 43046721, 1575, 6075, 32805, 129140163, 387420489, 2835, 10125, 98415

Offset: 1

Ray Chandler, Sep 22 2006

n-th row contains A001055(n) terms.

First item of each row gives A038547.

			Table begins:
  1,
  3,
  9,
  15, 27,
  81,
  45, 243,
  729,
  105, 135, 2187,
  ...

Amiram Eldar, Table of n, a(n) for n = 1..7314 (rows n=1..1000, flattened)

Cf. A001055, A038547, A077569, A162247.

row[n_] := Module[{e = f[n] - 1}, Sort[Times @@ (Prime[Range[2, Length[#]+1]]^Reverse[#]) & /@ e]]; Table[row[n], {n, 1, 25}] // Flatten (* Amiram Eldar, Jun 28 2025 using the function f by T. D. Noe at A162247 *)

A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 2, 5, 28, 277, 3985, 76117, 1833187, 53756682, 1871041538, 75809298105, 3521419837339, 185235838688677, 10923147890901151, 715989783027216302, 51793686238309903860, 4109310551278549543317, 355667047514571431358297, 33422937748872646130124797

Offset: 0

Gus Wiseman, Jul 17 2018

nonn

Note that all connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} are strict except for (123...n)(123...n).

			The a(2) = 5 connected multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (12)(12), (1)(2)(12). The multiset partitions (11)(22), (1)(1)(22), (2)(2)(11), (1)(1)(2)(2) are not connected.

Cf. A001055, A007716, A007717, A007719, A020555, A045778, A061742, A094574, A162247, A316974.

nn=10;
ser=Exp[-3/2+Exp[x]/2]*Sum[Exp[Binomial[n+1,2]*x]/n!,{n,0,3*nn}];
Round/@(CoefficientList[Series[1+Log[ser],{x,0,nn}],x]*Array[Factorial,nn+1,0]) (* based on Jean-François Alcover after Vladeta Jovovic *)
(*second program *)
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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]]]];
Length/@Table[Select[mps[Ceiling[Range[1/2,n,1/2]]],Length[csm[#]]==1&],{n,4}]

Logarithmic transform of A020555.

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 5, 1, 2, 2, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 3, 4, 2, 2, 1, 3, 2, 2, 1, 5, 1, 2, 3, 3, 2, 2, 1, 4, 3, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 3, 5, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

Also the number of distinct possible GCDs of factorizations of n into factors > 1.
Also the number of nonzero terms in row n of A317748.
a(prime^n) = A008619(n).
If n is squarefree and composite, a(n) = 2.

			The divisors of 36 that are possible GCDs of factorizations of 36 are {1, 2, 3, 6, 36}, so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000837, A001055, A014963, A045778, A050370, A162247, A281116, A289509.

Cf. A317748, A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[Length[Select[Divisors[n],goc[n,#]!=0&]],{n,100}]

A317751aux(n, m, facs, gcds) = if(1==n, setunion([gcd(Vec(facs))],gcds), my(newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); gcds = setunion(gcds,A317751aux(n/d, d, newfacs, gcds)))); (gcds));
A317751(n) = if(1==n,0,length(A317751aux(n, n, List([]), Set([])))); \\ Antti Karttunen, Sep 08 2018

More terms from Antti Karttunen, Sep 08 2018

A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Sep 15 2018

From Tian Vlasic, Dec 31 2021: (Start)
Numbers that have an equal number of even and odd-length unordered factorizations.
There are infinitely many terms since p^2 is a term for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures. (End)

			12 = 2*6 = 3*4 = 2*2*3 has an equal number of even-length factorizations and odd-length factorizations (2). - _Tian Vlasic_, Dec 09 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A319239.

Cf. A001055, A025487, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319238.

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}],0]

A317146 Moebius function in the ranked poset of factorizations of n into factors > 1, evaluated at the minimum (the prime factorization of n).

Original entry on oeis.org

0, 1, 1, -1, 1, -1, 1, 0, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 0, 1, 1, 2, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 1, 0, -1, 2, 1, 1, -1, 2, 1, 2, 1, -1, 1, 1

Offset: 1

Gus Wiseman, Jul 22 2018

sign

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

			The factorizations of 60 followed by their Moebius values are the following. The second column sums to 0, as required.
  (2*2*3*5) -> -3
   (2*2*15) ->  1
   (2*3*10) ->  2
    (2*5*6) ->  2
     (2*30) -> -1
    (3*4*5) ->  2
     (3*20) -> -1
     (4*15) -> -1
     (5*12) -> -1
     (6*10) -> -1
       (60) ->  1

Cf. A000837, A001055, A007716, A045778, A162247, A275024, A281113, A299202, A317144, A317145.

Product_{k>=2} 1/(1-a(n)/n^s) = 1+P(s), Re(s)>1, where P(s) is the prime zeta function. - Tian Vlasic, Jan 25 2024

A317176 Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 18, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 49, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 74, 1, 4, 1, 3, 1, 4, 1, 78, 1, 1, 3, 3, 1, 4, 1, 49, 6, 1, 1

Offset: 1

Gus Wiseman, Jul 23 2018

nonn

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

			The a(24) = 11 chains:
  (2*2*2*3) < (24)
  (2*2*2*3) < (2*12)  < (24)
  (2*2*2*3) < (3*8)   < (24)
  (2*2*2*3) < (4*6)   < (24)
  (2*2*2*3) < (2*2*6) < (24)
  (2*2*2*3) < (2*3*4) < (24)
  (2*2*2*3) < (2*2*6) < (2*12) < (24)
  (2*2*2*3) < (2*2*6) < (4*6)  < (24)
  (2*2*2*3) < (2*3*4) < (2*12) < (24)
  (2*2*2*3) < (2*3*4) < (3*8)  < (24)
  (2*2*2*3) < (2*3*4) < (4*6)  < (24)

Cf. A001055, A002846, A007716, A045778, A162247, A213427, A275024, A281113, A299202, A317144, A317145, A317146.

a(prime^n) = A213427(n).

A321460 Expansion of Product_{k>0} (1 - x^k)^A001055(k).

Original entry on oeis.org

1, -1, -1, 0, -1, 2, 0, 2, -1, 0, 3, -1, -2, -1, 1, -6, -1, 0, 0, 0, 7, -1, 1, -2, 4, 1, -2, 11, 1, -2, -10, 11, -12, 16, -15, -6, -6, -12, -1, 8, -4, -10, 9, -19, 21, -15, 23, 4, 28, -8, 42, -6, 9, 19, 3, -21, -18, -14, -15, 3, -72, 70, -21, -49, -9, 18, -12, 26, -68, -12

Offset: 0

Seiichi Manyama, Nov 10 2018

sign

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

Convolution inverse of A066739.

Cf. A001055, A066806, A162247, A321376, A321377.

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

cp's OEIS Frontend

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

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A372053 Irregular array read by rows: row n lists the factorizations of n into a product of nondecreasing integers >= 2.

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A122819 Array read by rows in which the n-th row contains smallest odd numbers in increasing order of all possible prime signatures with n divisors.

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A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

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A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

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A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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A317146 Moebius function in the ranked poset of factorizations of n into factors > 1, evaluated at the minimum (the prime factorization of n).

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A317176 Number of chains of factorizations of n into factors > 1, ordered by refinement, starting with the prime factorization of n and ending with the maximum factorization (n).

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A321460 Expansion of Product_{k>0} (1 - x^k)^A001055(k).