A330936 -id:A330936 - cp's OEIS frontend

A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

1, 4, 8, 12, 16, 24, 36, 48, 60, 72, 96, 120, 128, 144, 180, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1440, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456, 3600

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

This is the sorted list of positions of first appearances in A001055 of each element of the range (A045782).

			Factorizations of n for n = 4, 8, 12, 16, 24, 36, 48, 60:
  4    8      12     16       24       36       48         60
  2*2  2*4    2*6    2*8      3*8      4*9      6*8        2*30
       2*2*2  3*4    4*4      4*6      6*6      2*24       3*20
              2*2*3  2*2*4    2*12     2*18     3*16       4*15
                     2*2*2*2  2*2*6    3*12     4*12       5*12
                              2*3*4    2*2*9    2*3*8      6*10
                              2*2*2*3  2*3*6    2*4*6      2*5*6
                                       3*3*4    3*4*4      3*4*5
                                       2*2*3*3  2*2*12     2*2*15
                                                2*2*2*6    2*3*10
                                                2*2*3*4    2*2*3*5
                                                2*2*2*2*3

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

All terms belong to A025487
Includes all highly factorable numbers A033833.
Factorizations are A001055, with image A045782.
The least number with A045782(n) factorizations is A045783(n).
The least number with n factorizations is A330973(n).
The strict version is A330997.
Cf. A001222, A002033, A007716, A045778, A318284, A325238, A330935, A330936, A330976, A330977, A330989, A330991, A330992.

nn=1000;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[facs,nn];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

A045782 Number of factorizations of n for some n (image of A001055).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 15, 16, 19, 21, 22, 26, 29, 30, 31, 36, 38, 42, 45, 47, 52, 56, 57, 64, 66, 67, 74, 77, 92, 97, 98, 101, 105, 109, 118, 135, 137, 139, 141, 162, 165, 171, 176, 181, 189, 195, 198, 203, 212, 231, 249, 250, 254, 257, 267, 269, 272, 289

Offset: 1

David W. Wilson

nonn

Also the image of A318284. - Gus Wiseman, Jan 11 2020

Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.

Factorizations are A001055 with image this sequence and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with exactly a(n) factorizations is A045783(n).
The least number with exactly n factorizations is A330973(n).
Cf. A002033, A007716, A033833, A318284, A325238, A330935, A330936, A330977, A330989, A330991, A330992, A330997.

terms = 61; m0 = 10^5; dm = 10^4;
f[1, ] = 1; f[n, k_] := f[n, k] = Sum[f[n/d, d], {d, Select[Divisors[n], 1 < # <= k &]}];
Clear[seq]; seq[m_] := seq[m] = Sort[Tally[Table[f[n, n], {n, 1, m}]][[All, 1]]][[1 ;; terms]]; seq[m = m0]; seq[m += dm]; While[Print[m]; seq[m] != seq[m - dm], m += dm];
seq[m] (* Jean-François Alcover, Oct 04 2018 *)

The Luca et al. paper shows that the number of terms with a(n) <= x is x^{ O( log log log x / log log x )}. - N. J. A. Sloane, Jun 12 2009

Name edited by Gus Wiseman, Jan 11 2020

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 04 2020

This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.

			Triangle begins:
   1:          16: 0 1 3 2    31: 1            46: 0 1
   2: 1        17: 1          32: 0 1 5 8 4    47: 1
   3: 1        18: 0 1 2      33: 0 1          48: 0 1 10 23 15
   4: 0 1      19: 1          34: 0 1          49: 0 1
   5: 1        20: 0 1 2      35: 0 1          50: 0 1 2
   6: 0 1      21: 0 1        36: 0 1 7 7      51: 0 1
   7: 1        22: 0 1        37: 1            52: 0 1 2
   8: 0 1 1    23: 1          38: 0 1          53: 1
   9: 0 1      24: 0 1 5 5    39: 0 1          54: 0 1 5 5
  10: 0 1      25: 0 1        40: 0 1 5 5      55: 0 1
  11: 1        26: 0 1        41: 1            56: 0 1 5 5
  12: 0 1 2    27: 0 1 1      42: 0 1 3        57: 0 1
  13: 1        28: 0 1 2      43: 1            58: 0 1
  14: 0 1      29: 1          44: 0 1 2        59: 1
  15: 0 1      30: 0 1 3      45: 0 1 2        60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
  ()  (6*8)      (2*3*8)->(6*8)       (2*2*2*6)->(2*4*6)->(6*8)
      (2*24)     (2*4*6)->(6*8)       (2*2*3*4)->(2*3*8)->(6*8)
      (3*16)     (2*3*8)->(2*24)      (2*2*3*4)->(2*4*6)->(6*8)
      (4*12)     (2*3*8)->(3*16)      (2*2*2*6)->(2*4*6)->(2*24)
      (2*3*8)    (2*4*6)->(2*24)      (2*2*2*6)->(2*4*6)->(4*12)
      (2*4*6)    (2*4*6)->(4*12)      (2*2*3*4)->(2*3*8)->(2*24)
      (3*4*4)    (3*4*4)->(3*16)      (2*2*3*4)->(2*3*8)->(3*16)
      (2*2*12)   (3*4*4)->(4*12)      (2*2*3*4)->(2*4*6)->(2*24)
      (2*2*2*6)  (2*2*12)->(2*24)     (2*2*3*4)->(2*4*6)->(4*12)
      (2*2*3*4)  (2*2*12)->(4*12)     (2*2*3*4)->(3*4*4)->(3*16)
                 (2*2*2*6)->(6*8)     (2*2*3*4)->(3*4*4)->(4*12)
                 (2*2*3*4)->(6*8)     (2*2*2*6)->(2*2*12)->(2*24)
                 (2*2*2*6)->(2*24)    (2*2*2*6)->(2*2*12)->(4*12)
                 (2*2*2*6)->(4*12)    (2*2*3*4)->(2*2*12)->(2*24)
                 (2*2*3*4)->(2*24)    (2*2*3*4)->(2*2*12)->(4*12)
                 (2*2*3*4)->(3*16)
                 (2*2*3*4)->(4*12)
                 (2*2*2*6)->(2*4*6)
                 (2*2*3*4)->(2*3*8)
                 (2*2*3*4)->(2*4*6)
                 (2*2*3*4)->(3*4*4)
                 (2*2*2*6)->(2*2*12)
                 (2*2*3*4)->(2*2*12)

Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upfacs[y],y],{y,facs[n]}],{n},First[facs[n]]],Length[#]==k-1&]],{n,100},{k,PrimeOmega[n]}]

T(2^n,k) = A330785(n,k).

T(n,1) + T(n,2) = 1.

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A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

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A045782 Number of factorizations of n for some n (image of A001055).

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A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

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