A340654 -id:A340654 - cp's OEIS frontend

A106529 Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

2, 6, 9, 20, 30, 45, 50, 56, 75, 84, 125, 126, 140, 176, 189, 196, 210, 264, 294, 315, 350, 396, 416, 440, 441, 490, 525, 594, 616, 624, 660, 686, 735, 875, 891, 924, 936, 968, 990, 1029, 1040, 1088, 1100, 1225, 1386, 1404, 1452, 1456, 1485, 1540, 1560

Offset: 1

Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005

nonn

It seems that the ratio between successive terms tends to 1 as n increases, meaning perhaps that most numbers are in this sequence.
The number of terms that have the k-th prime as their largest prime factor is A000984(k), the k-th central binomial coefficient. E.g., 6 and 9 are the A000984(2)=2 terms in {a(n)} that have prime(2)=3 as their largest prime factor.
The sequence contains the positive integers m such that the rank of the partition B(m) = 0. For m >= 2, B(m) is defined as the partition obtained by taking the prime decomposition of m and replacing each prime factor p with its index i (i.e., i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. The rank of a partition P is the largest part of P minus the number of parts of P. - Emeric Deutsch, May 09 2015
Also Heinz numbers of balanced partitions, counted by A047993. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Feb 08 2021

			a(7)=50 because 50=2*5*5 is, for k=3, the product of k primes, the largest of which is the k-th prime, and 50 is the 7th such number.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000984.
A001222 counts prime factors.
A056239 adds up prime indices.
A061395 selects maximum prime index.
A112798 lists the prime indices of each positive integer.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A090858 counts partitions of rank 1.
- A098124 counts balanced compositions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A006141, A064174, A096401, A117409, A168659, A324522, A340654, A340655, A340656, A340657.

with(numtheory): a := proc (n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: A := {}: for i from 2 to 1600 do if a(i) = 0 then A := `union`(A, {i}) else  end if end do: A; # Emeric Deutsch, May 09 2015

Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)

For all terms, A001222(a(n)) = A061395(a(n)). - Gus Wiseman, Feb 08 2021

A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485

Offset: 1

N. J. A. Sloane

nonn

Or, number of partitions of n in which number of largest parts is equal to the largest part.
a(n) is the number of partitions of n-1 without parts that differ by less than 2 and which have no parts less than three. [MacMahon]
There are two conflicting choices for the offset in this sequence. For the definition given here the offset is 1, and that is what we shall adopt. On the other hand, if one arrives at this sequence via the Rogers-Ramanujan identities (see the next comment), the natural offset is 0.
Related to Rogers-Ramanujan identities: Let G[1](q) and G[2](q) be the generating functions for the two Rogers-Ramanujan identities of A003114 and A003106, starting with the constant term 1. The g.f. for the present sequence is G[3](q) = (G[1](q) - G[2](q))/q = 1+q^3+q^4+q^5+q^6+q^7+2*q^8+2*q^9+3*q^10+.... - Joerg Arndt, Oct 08 2012; N. J. A. Sloane, Nov 18 2015
For more about the generalized Rogers-Ramanujan series G[i](x) see the Andrews-Baxter and Lepowsky-Zhu papers. The present series is G[3](x). - N. J. A. Sloane, Nov 22 2015
From Wolfdieter Lang, Oct 31 2016: (Start)
From Hardy (H) p. 94, eq. (6.12.1) and Hardy-Wright (H-W), p. 293,  eq. (19.14.3) for H_2(a,x) - H_1(a,x) = a*H_1(a*x,x) one finds from the result for H_1(a,x) (in (H) on top on p. 95), after putting a=x, the o.g.f. of a(n) = A003114(n) - A003106(n), n >= 0, with a(0) = 0 as Sum_{m>=0} x^((m+1)^2) / Product_{j=1..m} (1 - x^j). The m=0 term is  1*x^1. See the formula given by Joerg Arndt, Jan 29 2011.
This formula has a combinatorial interpretation (found similar to the one given in (H) section 6.0, pp. 91-92 or (H-W) pp. 290-291): a(n) is the number of partitions of n with parts differing by at least 2 and part 1 present. See the example for a(15) below. (End)
The Heinz numbers of these integer partitions are given by A324522. - Gus Wiseman, Mar 09 2019

			G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
3 + 6 + 6,
3 + 5 + 7,
3 + 4 + 8,
3 + 3 + 9, and
2 + 13.
- _Joerg Arndt_, Oct 08 2012
a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and  [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - _Wolfdieter Lang_, Oct 31 2016
a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index  m) and  m*(m+2) = 3 + 5 + ... + (2*m+1). - _Wolfdieter Lang_, Nov 02 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(8) = 1 through a(15) = 5 integer partitions:
  (6,2)  (7,2)    (8,2)    (9,2)    (10,2)   (11,2)   (12,2)   (13,2)
         (3,3,3)  (4,3,3)  (4,4,3)  (5,4,3)  (5,5,3)  (6,5,3)  (6,6,3)
                           (5,3,3)  (6,3,3)  (6,4,3)  (7,4,3)  (7,5,3)
                                             (7,3,3)  (8,3,3)  (8,4,3)
                                                               (9,3,3)
(End)

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
George E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see A003114, A003106, A006141, A264591, A264592, A264593, A264594, A264595. G[0] = G[1]+G[2] is given by A003113.
Cf. A064174, A090858, A324516, A324518, A324520, A324522.
A003106 counts partitions with minimum > length.
A003114 counts partitions with minimum >= length.
A026794 counts partitions by minimum.
A039899 counts partitions with minimum < length.
A039900 counts partitions with minimum <= length.
A239950 counts partitions with minimum equal to number of distinct parts.
Sequences related to balance:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 ranks balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A055396, A114638, A117409, A325134, A340601, A340611, A340654, A340655.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i,i)))))
    end:
a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 08 2012

b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Min[#]==Length[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)

{a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos, Jan 22 2008 */

G.f.: Sum_{m>=1} (x^(m^2)-x^(m*(m+1))) / Product_{i=1..m} (1-x^i) .
G.f.: Sum_{n>=1} x^(n^2)/Product_{k=1..n-1} (1-x^k). - Joerg Arndt, Jan 29 2011
a(n) = A003114(n) - A003106(n) = A039900(n) - A039899(n), (offset 1). - Vladeta Jovovic, Jul 17 2004
Plouffe in his 1992 dissertation conjectured that this has g.f. = (1+z+z^4+2*z^5-z^3-z^8+3*z^10-z^7+z^9)/(1+z-z^4-2*z^3-z^8+z^10), but Michael Somos pointed out on Jan 22 2008 that this is false.
Expansion of ( f(-x^2, -x^3) - f(-x, -x^4) ) / f(-x) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 22 2007
a(n) ~ sqrt(1/sqrt(5) - 2/5) * exp(2*Pi*sqrt(n/15)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 01 2016

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
Better description from Naohiro Nomoto, Feb 06 2002
Name shortened by Gus Wiseman, Apr 07 2021 (balanced partitions are A047993).

A340653 Number of balanced factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

			The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
  3*5*8    2*8*9    3*8*8      4*8*9      6*8*9      8*8*12
  2*2*30   3*6*8    4*6*8      6*6*8      2*8*27     2*2*8*24
  2*3*20   2*4*18   2*8*12     2*8*18     3*8*18     2*3*8*16
  2*5*12   2*6*12   4*4*12     3*8*12     4*4*27     2*4*4*24
           3*4*12   2*2*2*24   4*4*18     4*6*18     2*4*6*16
                    2*2*3*16   4*6*12     4*9*12     3*4*4*16
                               2*12*12    6*6*12     2*2*12*16
                               2*2*2*36   2*12*18    2*2*2*2*48
                               2*2*3*24   3*12*12    2*2*2*3*32
                               2*3*3*16   2*2*2*54
                                          2*2*3*36
                                          2*3*3*24
                                          3*3*3*16

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

Positions of zeros are A001358.
Positions of nonzero terms are A100959.
The co-balanced version is A340596.
Taking maximum factor instead of maximum Omega gives A340599.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.

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],#=={}||Length[#]==Max[PrimeOmega/@#]&]],{n,100}]

A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ Antti Karttunen, Oct 22 2023

Data section extended up to a(120) by Antti Karttunen, Oct 22 2023

A340596 Number of co-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

			The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
  2*6    3*8     4*9     8*9     3*5*8     2*72     4*5*9
  3*4    4*6     6*6     2*36    4*5*6     3*48     5*6*6
         2*12    2*18    3*24    2*2*30    4*36     2*2*45
                 3*12    4*18    2*3*20    6*24     2*3*30
                         6*12    2*4*15    8*18     2*5*18
                                 2*5*12    9*16     2*6*15
                                 2*6*10    12*12    2*9*10
                                 3*4*10             3*3*20
                                                    3*4*15
                                                    3*5*12
                                                    3*6*10

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The version for unlabeled multiset partitions is A319616.
The alt-balanced version is A340599.
The balanced version is A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

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],Length[#]==PrimeNu[n]&]],{n,100}]

A340596(n, m=n, om=omega(n)) = if(1==n,(0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ Antti Karttunen, Jun 10 2024

Data section extended up to a(120) by Antti Karttunen, Jun 10 2024

A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 1

Offset: 1

Gus Wiseman, Jan 20 2021

nonn

I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021

			The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
  3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
  2*2*2*2*2*6   2*2*3*4*6*6   2*3*4*4*6*6         2*2*2*2*2*6*6*8
                2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
                              2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
                              2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
                                                  2*2*2*2*2*2*2*8*9
                                                  2*2*2*2*2*2*4*4*9

Antti Karttunen, Table of n, a(n) for n = 1..100000

The co-balanced version is A340596.
Positions of nonzero terms are A340597.
The case of powers of two is A340611.
Taking maximum Omega instead of maximum factor gives A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

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],Length[#]==Max[#]&]],{n,100}]

A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024

A340655 Number of twice-balanced factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15

The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340652.
The balanced version is A340653.
The cross-balanced version is A340654.
Positions of zeros are A340656.
Positions of nonzero terms are A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, A340608.

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],#=={}||Length[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 104, 106, 116, 118, 122, 124, 128, 134, 136, 142, 146, 148, 152, 158, 164, 166, 172, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 232, 236, 244

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
      1: {}              44: {1,1,5}          106: {1,16}
      2: {1}             46: {1,9}            116: {1,1,10}
      4: {1,1}           52: {1,1,6}          118: {1,17}
      6: {1,2}           58: {1,10}           122: {1,18}
      8: {1,1,1}         62: {1,11}           124: {1,1,11}
     10: {1,3}           64: {1,1,1,1,1,1}    128: {1,1,1,1,1,1,1}
     14: {1,4}           68: {1,1,7}          134: {1,19}
     16: {1,1,1,1}       74: {1,12}           136: {1,1,1,7}
     20: {1,1,3}         76: {1,1,8}          142: {1,20}
     22: {1,5}           82: {1,13}           146: {1,21}
     26: {1,6}           86: {1,14}           148: {1,1,12}
     28: {1,1,4}         88: {1,1,1,5}        152: {1,1,1,8}
     32: {1,1,1,1,1}     92: {1,1,9}          158: {1,22}
     34: {1,7}           94: {1,15}           164: {1,1,13}
     38: {1,8}          104: {1,1,1,6}        166: {1,23}
For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.

The version looking at greatest factor is A000079.
The version for twice-balanced is A340656, with complement A340657.
These factorization are counted by A340832.
The complement is A340855.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors.
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.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

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

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

Original entry on oeis.org

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/#&]!={})]]

A340597 Numbers with an alt-balanced factorization.

Original entry on oeis.org

4, 12, 18, 27, 32, 48, 64, 72, 80, 96, 108, 120, 128, 144, 160, 180, 192, 200, 240, 256, 270, 288, 300, 320, 360, 384, 400, 405, 432, 448, 450, 480, 500, 540, 576, 600, 640, 648, 672, 675, 720, 750, 768, 800, 864, 896, 900, 960, 972, 1000, 1008, 1024, 1080

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization into factors > 1 to be alt-balanced if its length is equal to its greatest factor.

			The sequence of terms together with their prime signatures begins:
      4: (2)        180: (2,2,1)    450: (1,2,2)
     12: (2,1)      192: (6,1)      480: (5,1,1)
     18: (1,2)      200: (3,2)      500: (2,3)
     27: (3)        240: (4,1,1)    540: (2,3,1)
     32: (5)        256: (8)        576: (6,2)
     48: (4,1)      270: (1,3,1)    600: (3,1,2)
     64: (6)        288: (5,2)      640: (7,1)
     72: (3,2)      300: (2,1,2)    648: (3,4)
     80: (4,1)      320: (6,1)      672: (5,1,1)
     96: (5,1)      360: (3,2,1)    675: (3,2)
    108: (2,3)      384: (7,1)      720: (4,2,1)
    120: (3,1,1)    400: (4,2)      750: (1,1,3)
    128: (7)        405: (4,1)      768: (8,1)
    144: (4,2)      432: (4,3)      800: (5,2)
    160: (5,1)      448: (6,1)      864: (5,3)
For example, there are two alt-balanced factorizations of 480, namely (2*3*4*4*5) and (2*2*2*2*5*6), so 480 in the sequence.

Numbers with a balanced factorization are A100959.
These factorizations are counted by A340599.
The twice-balanced version is A340657.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
- A340655 counts twice-balanced factorizations.
Cf. A006141, A064174, A117409, A200750, A303975, A324518, A324522, A325134, A340607, A340608, A340611, A340656.

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

A340656 Numbers without a twice-balanced factorization.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 42, 46, 48, 49, 51, 55, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 102, 105, 106, 108, 110, 111, 112, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jan 16 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The sequence of terms together with their prime indices begins:
     4: {1,1}          33: {2,5}          64: {1,1,1,1,1,1}
     6: {1,2}          34: {1,7}          65: {3,6}
     8: {1,1,1}        35: {3,4}          66: {1,2,5}
     9: {2,2}          38: {1,8}          69: {2,9}
    10: {1,3}          39: {2,6}          70: {1,3,4}
    14: {1,4}          42: {1,2,4}        72: {1,1,1,2,2}
    15: {2,3}          46: {1,9}          74: {1,12}
    16: {1,1,1,1}      48: {1,1,1,1,2}    77: {4,5}
    21: {2,4}          49: {4,4}          78: {1,2,6}
    22: {1,5}          51: {2,7}          80: {1,1,1,1,3}
    25: {3,3}          55: {3,5}          81: {2,2,2,2}
    26: {1,6}          57: {2,8}          82: {1,13}
    27: {2,2,2}        58: {1,10}         84: {1,1,2,4}
    30: {1,2,3}        60: {1,1,2,3}      85: {3,7}
    32: {1,1,1,1,1}    62: {1,11}         86: {1,14}
For example, the factorizations of 48 with (2) and (3) equal are: (2*2*2*6), (2*2*3*4), (2*4*6), (3*4*4), but since none of these has length 2, the sequence contains 48.

Positions of zeros in A340655.
The complement is A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.

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

cp's OEIS Frontend

A106529 Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.

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A006141 Number of integer partitions of n whose smallest part is equal to the number of parts.

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A340653 Number of balanced factorizations of n.

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A340596 Number of co-balanced factorizations of n.

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A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

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A340655 Number of twice-balanced factorizations of n.

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A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

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