Displaying 1-10 of 29 results found.
Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.
+10
109
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
COMMENTS
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
EXAMPLE
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.
MAPLE
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
MATHEMATICA
Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)
CROSSREFS
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.
AUTHOR
Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005
Number of integer partitions of n whose smallest part is equal to the number of parts.
(Formerly M0260)
+10
67
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
COMMENTS
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 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
REFERENCES
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).
FORMULA
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 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
EXAMPLE
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.
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 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)
MAPLE
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)):
MATHEMATICA
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-Fran�ois Alcover, Mar 17 2014, after Alois P. Heinz *) Table[Length[Select[IntegerPartitions[n], Min[#]==Length[#]&]], {n, 30}] (* Gus Wiseman, Mar 09 2019 *)
PROG
(PARI) {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 */
CROSSREFS
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. 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.
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
Number of balanced factorizations of n.
+10
42
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
COMMENTS
A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega ( A001222).
EXAMPLE
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
MATHEMATICA
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}]
PROG
(PARI) 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
CROSSREFS
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. 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.
Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.
+10
41
0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 2, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 1, 10, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 1, 0, 0, 0, 7, 4, 0, 0, 2, 0, 0
COMMENTS
For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).
EXAMPLE
The a(1) = 0 through a(24) = 3 factorizations:
... 2*2 ... 2*4 3*3 .. 2*2*3 ... 2*8 . 2*3*3 . 2*2*5 ... 2*2*6
2*2*2 4*4 2*3*4
2*2*4 2*2*2*3
2*2*2*2
MATHEMATICA
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[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]], {n, 100}]
CROSSREFS
The complement is counted by A368414.
Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.
+10
37
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2
COMMENTS
For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).
EXAMPLE
The a(n) factorizations for selected n:
1 6 12 24 30 60 72 120
2*3 2*6 2*12 2*15 2*30 2*36 2*60
3*4 3*8 3*10 3*20 3*24 3*40
4*6 5*6 4*15 4*18 4*30
2*3*5 5*12 6*12 5*24
6*10 8*9 6*20
2*3*10 8*15
2*5*6 10*12
3*4*5 2*3*20
2*5*12
2*6*10
3*4*10
3*5*8
4*5*6
MATHEMATICA
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[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]], {n, 100}]
CROSSREFS
The complement is counted by A368413.
Number of non-condensed integer factorizations of n into unordered factors > 1.
+10
31
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, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0
COMMENTS
A multiset is condensed iff it is possible to choose a different divisor of each element.
EXAMPLE
The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).
MATHEMATICA
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[Select[Tuples[Divisors /@ #], UnsameQ@@#&]]==0&]], {n, 100}]
CROSSREFS
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of condensed integer factorizations of n into unordered factors > 1.
+10
29
1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2
COMMENTS
A multiset is condensed iff it is possible to choose a different divisor of each element.
EXAMPLE
The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).
MATHEMATICA
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[Select[Tuples[Divisors /@ #], UnsameQ@@#&]]>0&]], {n, 100}]
CROSSREFS
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of factorizations of n into factors > 1 with length and greatest factor equal.
+10
24
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
COMMENTS
I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021
EXAMPLE
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
MATHEMATICA
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}]
PROG
(PARI) 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
CROSSREFS
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. 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.
EXTENSIONS
Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024
Number of cross-balanced factorizations of n.
+10
23
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, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 5, 1, 1, 1, 3, 1
COMMENTS
We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).
EXAMPLE
The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
2*6 4*6 4*9 2*4*9 4*4*9 8*30
3*4 2*2*6 6*6 2*6*6 4*6*6 12*20
2*3*4 2*2*9 3*4*6 2*2*4*9 5*6*8
2*3*6 2*2*2*9 2*2*6*6 2*4*30
3*3*4 2*2*3*6 2*3*4*6 2*6*20
2*3*3*4 3*3*4*4 2*8*15
2*2*2*2*9 3*4*20
2*2*2*3*6 3*8*10
2*2*3*3*4 4*5*12
2*10*12
2*3*5*8
2*2*2*30
2*2*3*20
2*2*5*12
MATHEMATICA
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], #=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]], {n, 100}]
PROG
(PARI) A340654(n, m=n, om=omega(n), mbo=0) = if(1==n, (mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo, bigomega(d)))))); \\ Antti Karttunen, Jun 19 2024
CROSSREFS
Positions of terms > 1 are A126706. The co-balanced version is A340596. The version for unlabeled multiset partitions is A340651. The twice-balanced version is A340655. 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. - A340599 counts alt-balanced factorizations. - A340652 counts unlabeled twice-balanced multiset partitions. - A340656 have no twice-balanced factorizations. - A340657 have a twice-balanced factorization.
Number of twice-balanced factorizations of n.
+10
22
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
COMMENTS
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;
EXAMPLE
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
MATHEMATICA
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}]
CROSSREFS
The co-balanced version is A340596. The version for unlabeled multiset partitions is A340652. The cross-balanced version is A340654. Positions of nonzero terms are A340657. 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.
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