A340101 -id:A340101 - cp's OEIS frontend

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
     3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
     5: {3}          29: {10}         49: {4,4}
     7: {4}          30: {1,2,3}      50: {1,3,3}
     9: {2,2}        31: {11}         51: {2,7}
    11: {5}          33: {2,5}        53: {16}
    12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
    13: {6}          36: {1,1,2,2}    55: {3,5}
    15: {2,3}        37: {12}         56: {1,1,1,4}
    17: {7}          39: {2,6}        57: {2,8}
    18: {1,2,2}      40: {1,1,1,3}    59: {17}
    19: {8}          41: {13}         60: {1,1,2,3}
    21: {2,4}        42: {1,2,4}      61: {18}
    23: {9}          43: {14}         63: {2,2,4}
    24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
    25: {3,3}        47: {15}         66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

The version looking at greatest factor is A057716.
The version for twice-balanced is A340657, with complement A340656.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors, listed by A161906.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]

A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 6, 3, 10, 7, 18, 15, 30, 28, 51, 50, 82, 87, 134, 145, 211, 235, 331, 375, 510, 586, 779, 901, 1172, 1366, 1750, 2045, 2581, 3026, 3778, 4433, 5476, 6430, 7878, 9246, 11240, 13189, 15931, 18670, 22417, 26242, 31349, 36646, 43567, 50854

Offset: 1

Gus Wiseman, Jan 08 2021

nonn

			The a(3) = 2 through a(10) = 7 partitions:
  3     5       321   7         332     9           532
  111   311           322       521     333         541
        11111         331       32111   522         721
                      511               531         32221
                      31111             711         33211
                      1111111           32211       52111
                                        33111       3211111
                                        51111
                                        3111111
                                        111111111

Partitions of odd length are counted by A027193, ranked by A026424.
Partitions with odd maximum are counted by A027193, ranked by A244991.
The Heinz numbers of these partitions are given by A340386.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions with odd rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A027187, A078408, A174725, A236914.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]*Max[#]]&]],{n,30}]

A340785 Number of factorizations of 2n into even factors > 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 4, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 7, 1, 2, 1, 4, 1, 3, 1, 11, 1, 2, 1, 6, 1, 2, 1, 7, 1, 3, 1, 4, 1, 2, 1, 12, 1, 3, 1, 4, 1, 3, 1, 7, 1, 2, 1, 7, 1, 2, 1, 15, 1, 3, 1, 4, 1, 3, 1, 12, 1, 2, 1, 4, 1, 3, 1, 12, 1, 2, 1, 7, 1

Offset: 1

Gus Wiseman, Jan 30 2021

nonn

			The a(n) factorizations for n = 2*2, 2*4, 2*8, 2*12, 2*16, 2*32, 2*36, 2*48 are:
  4    8      16       24     32         64           72      96
  2*2  2*4    2*8      4*6    4*8        8*8          2*36    2*48
       2*2*2  4*4      2*12   2*16       2*32         4*18    4*24
              2*2*4    2*2*6  2*2*8      4*16         6*12    6*16
              2*2*2*2         2*4*4      2*4*8        2*6*6   8*12
                              2*2*2*4    4*4*4        2*2*18  2*6*8
                              2*2*2*2*2  2*2*16               4*4*6
                                         2*2*2*8              2*2*24
                                         2*2*4*4              2*4*12
                                         2*2*2*2*4            2*2*4*6
                                         2*2*2*2*2*2          2*2*2*12
                                                              2*2*2*2*6

Antti Karttunen, Table of n, a(n) for n = 1..20000

Note: A-numbers of Heinz-number sequences are in parentheses below.
The version for partitions is A035363 (A066207).
The odd version is A340101.
The even length case is A340786.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
A316439 counts factorizations by product and length
A340102 counts odd-length factorizations of odd numbers into odd factors.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum.
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A050320, A066208, A160786, A174725, A320655, A320656, A339890, A340851.
Even bisection of A349906.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[#,OddQ]=={}&]],{n,2,100,2}]

A349906(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A349906(n/d, d))); (s));
A340785(n) = A349906(2*n); \\ Antti Karttunen, Dec 13 2021

a(n) = A349906(2*n). - Antti Karttunen, Dec 13 2021

A340832 Number of factorizations of n into factors > 1 with odd least factor.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 1, 1, 1, 1, 0, 4, 0, 1, 2, 2, 1, 2, 0, 1, 2, 2, 1, 2, 0, 1, 3, 1, 0, 4, 0, 2, 1, 1, 0, 2, 2, 1, 3, 1, 0, 4, 0, 2, 1, 1, 1, 5, 0, 1, 3, 2, 0, 2, 0, 1, 5, 2, 0, 2, 0, 2, 2, 1, 1, 4, 1, 1, 1, 1, 0, 5, 0, 1, 6

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)     (3*36)     (135)      (3*60)     (3*84)
  (5*9)    (9*12)     (3*45)     (5*36)     (7*36)
  (3*15)   (3*4*9)    (5*27)     (9*20)     (9*28)
  (3*3*5)  (3*6*6)    (9*15)     (5*6*6)    (3*3*28)
           (3*3*12)   (3*5*9)    (3*3*20)   (3*4*21)
           (3*3*3*4)  (3*3*15)   (3*4*15)   (3*6*14)
                      (3*3*3*5)  (3*5*12)   (3*7*12)
                                 (3*6*10)   (3*3*4*7)
                                 (3*3*4*5)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A340854.
Positions of nonzero terms are A340855.
The version for partitions is A026804.
Odd-length factorizations are counted by A339890.
The version looking at greatest factor is A340831.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340607 counts factorizations with odd length and greatest factor.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A026424 lists numbers with odd Omega.
A027193 counts partitions of odd length.
A058695 counts partitions of odd numbers (A300063).
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A244991 lists numbers whose greatest prime index is odd.
A340692 counts partitions of odd rank.
Cf. A050320, A160786, A340385, A340596, A340599, A340654, A340655, A340931.

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

A340832(n, m=n, fc=1) = if(1==n, (m%2)&&!fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A340832(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A342086 Number of strict factorizations of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 9, 2, 5, 5, 7, 2, 9, 2, 9, 5, 5, 2, 16, 3, 5, 5, 9, 2, 15, 2, 10, 5, 5, 5, 18, 2, 5, 5, 16, 2, 15, 2, 9, 9, 5, 2, 25, 3, 9, 5, 9, 2, 16, 5, 16, 5, 5, 2, 31, 2, 5, 9, 14, 5, 15, 2, 9, 5, 15, 2, 34, 2, 5, 9, 9, 5, 15, 2, 25, 7, 5

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

A strict factorization of n is a set of distinct positive integers > 1 with product n.

			The a(1) = 1 through a(12) = 9 factorizations:
  ()  ()   ()   ()   ()   ()     ()   ()     ()   ()     ()    ()
      (2)  (3)  (2)  (5)  (2)    (7)  (2)    (3)  (2)    (11)  (2)
                (4)       (3)         (4)    (9)  (5)          (3)
                          (6)         (8)         (10)         (4)
                          (2*3)       (2*4)       (2*5)        (6)
                                                               (12)
                                                               (2*3)
                                                               (2*6)
                                                               (3*4)

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

A version for partitions is A026906 (strict partitions of 1..n).
A version for partitions is A036469 (strict partitions of 0..n).
A version for partitions is A047966 (strict partitions of divisors).
The non-strict version is A057567.
A000005 counts divisors, with sum A000203.
A000009 counts strict partitions.
A001055 counts factorizations, with strict case A045778.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.

sf1:= proc(n,m)
  local D,d;
  if n = 1 then return 1 fi;
  D:= select(`<`,numtheory:-divisors(n) minus {1},m);
  add( procname(n/d,d), d= D)
end proc:
sf:= proc(n) option remember; sf1(n,n+1) end proc:f:= proc(n) local d; add(sf(d),d=numtheory:-divisors(n)) end proc:map(f, [$1..100]); # Robert Israel, Mar 10 2021

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Select[facs[k],UnsameQ@@#&]],{k,Divisors[n]}],{n,30}]

A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.

Original entry on oeis.org

1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also numbers that can be factored in such a way that the length is divisible by the least common multiple.

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
For example, 24576 has three suitable factorizations:
  (2*2*2*2*2*2*2*2*2*2*2*12)
  (2*2*2*2*2*2*2*2*2*2*4*6)
  (2*2*2*2*2*2*2*2*2*3*4*4)
so is in the sequence.

Partitions of this type are counted by A340693 (A340606).
These factorizations are counted by A340851.
The reciprocal version is A340853.
A143773 counts partitions whose parts are multiples of the number of parts.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
A340785 counts factorizations into even numbers, even-length case A340786.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],And@@IntegerQ/@(Length[#]/#)&]!={}&]

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 2, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 1, 4, 1, 1, 4, 0, 2, 3, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 3, 1, 3, 5, 1, 1, 5, 2, 1, 2, 3, 1, 5, 2, 2, 2, 1, 2, 1, 1, 2, 4, 4, 1, 3, 1, 3, 5, 1, 1, 6

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

			The a(n) factorizations for n = 45, 108, 135, 180, 252:
  (45)      (4*27)        (135)       (4*45)        (4*63)
  (5*9)     (2*6*9)       (3*45)      (12*15)       (12*21)
  (3*15)    (3*4*9)       (5*27)      (4*5*9)       (4*7*9)
  (3*3*5)   (2*2*27)      (9*15)      (2*2*45)      (6*6*7)
            (2*2*3*9)     (3*5*9)     (2*6*15)      (2*2*63)
            (2*2*3*3*3)   (3*3*15)    (3*4*15)      (2*6*21)
                          (3*3*3*5)   (2*2*5*9)     (3*4*21)
                                      (3*3*4*5)     (2*2*7*9)
                                      (2*2*3*15)    (2*3*6*7)
                                      (2*2*3*3*5)   (3*3*4*7)
                                                    (2*2*3*21)
                                                    (2*2*3*3*7)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.

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

A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021

A340786 Number of factorizations of 4n into an even number of even factors > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jan 31 2021

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
  4*6   6*8      2*48      2*72      4*60      4*80          40*50
  2*12  2*24     4*24      4*36      6*40      8*40          4*500
        4*12     6*16      6*24      8*30      10*32         8*250
        2*2*2*6  8*12      8*18      10*24     16*20         10*200
                 2*2*4*6   12*12     12*20     2*160         20*100
                 2*2*2*12  2*2*6*6   2*120     2*2*2*40      2*1000
                           2*2*2*18  2*2*2*30  2*2*4*20      2*2*10*50
                                     2*2*6*10  2*2*8*10      2*2*2*250
                                               2*4*4*10      2*10*10*10
                                               2*2*2*2*2*10

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

Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.

g:= proc(n, m, p)
 option remember;
 local F,r,x,i;
 # number of factorizations of n into even factors > m with number of factors == p (mod 2)
 if n = 1 then if p = 0 then return 1 else return 0 fi fi;
 if m > n  or n::odd then return 0 fi;
 F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)),list));
 r:= 0;
 for x in F do
   for i from 1 while n mod x^i = 0 do
     r:= r + procname(n/x^i, x, (p+i) mod 2)
 od od;
 r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n],EvenQ[Length[#]]&&Select[#,OddQ]=={}&]],{n,100}]

A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022

A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose number of factors is divisible by their least common multiple.

			The a(n) factorizations for n = 8192, 46656, 73728:
  2*2*2*2*2*4*8*8          6*6*6*6*6*6              2*2*2*2*2*2*2*2*2*4*6*6
  2*2*2*2*4*4*4*8          2*2*2*2*2*2*3*3*3*3*3*3  2*2*2*2*2*2*2*2*3*4*4*6
  2*2*2*4*4*4*4*4                                   2*2*2*2*2*2*2*3*3*4*4*4
  2*2*2*2*2*2*2*2*2*2*2*4                           2*2*2*2*2*2*2*2*2*2*6*12
                                                    2*2*2*2*2*2*2*2*2*3*4*12

The version for partitions is A340693, with reciprocal version A143773.
Positions of nonzero terms are A340852.
The reciprocal version is A340853.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
A340831/A340832 count factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

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

cp's OEIS Frontend

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

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A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

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A340785 Number of factorizations of 2n into even factors > 1.

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A340832 Number of factorizations of n into factors > 1 with odd least factor.

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A342086 Number of strict factorizations of divisors of n.

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A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.

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A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

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A340831 Number of factorizations of n into factors > 1 with odd greatest factor.

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