A050326 -id:A050326 - cp's OEIS frontend

A382200 Numbers that can be written as a product of distinct squarefree numbers.

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Gus Wiseman, Mar 21 2025

nonn

First differs from A339741 in having 1080.
First differs from A382075 in having 18000.
These are positions of positive terms in A050326, complement A293243.
Also numbers whose prime indices can be partitioned into distinct sets.
Differs from A212167, which does not include 18000 = 2^4*3^2*5^3, for example. - R. J. Mathar, Mar 23 2025

			The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets, so 1080 is in the sequence.
We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.

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

Twice-partitions of this type are counted by A279785, see also A358914.
Normal multisets not of this type are counted by A292432, strong A292444.
The complement is A293243, counted by A050342.
The case of a unique choice is A293511.
MM-numbers of multiset partitions into distinct sets are A302494.
For distinct block-sums instead of blocks we have A382075, counted by A381992.
Partitions of this type are counted by A382077, complement A382078.
Normal multisets of this type are counted by A382214, strong A381996.
A001055 counts multiset partitions of prime indices, strict A045778.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A001222, A005117, A050345, A089259, A116539, A270995, A381441, A382201, A382216.

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
remove(t -> A[t]=0, [$1..N]); # Robert Israel, Apr 21 2025

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[sqfacs[#]]>0&]

A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes.

			The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
  (6)    (5*6)    (6*10)    (6*35)     (2*6*35)
  (2*3)  (2*15)   (2*5*6)   (10*21)    (5*6*14)
         (3*10)   (2*3*10)  (14*15)    (6*7*10)
         (2*3*5)            (5*6*7)    (2*10*21)
                            (2*3*35)   (2*14*15)
                            (2*5*21)   (2*5*6*7)
                            (2*7*15)   (3*10*14)
                            (3*5*14)   (2*3*5*14)
                            (3*7*10)   (2*3*7*10)
                            (2*3*5*7)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008966 with A339661.
A008966 allows only primes.
A339661 does not allow primes, only squarefree semiprimes.
A339740 lists the positions of zeros.
A339741 lists the positions of positive terms.
A339839 allows nonsquarefree semiprimes.
A339887 is the non-strict version.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339840 cannot be factored into distinct primes or semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050320 into squarefree numbers.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339742 [this sequence] into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A058696 counts all partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A339656 counts loop-graphical partitions (A339658).
-
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 has no additional conditions (A028260).
- A338914/A339562 can be partitioned into edges (A320911).
- A338916/A339563 can be partitioned into distinct pairs (A320912).
- A339559/A339564 cannot be partitioned into distinct edges (A320894).
- A339560/A339619 can be partitioned into distinct edges (A339561).
Cf. A000070, A001221, A005117, A320893, A320923, A338899, A339113, A339617, A353471.

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[sqps[n]],{n,100}]

A353471(n) = (numdiv(n)==2*omega(n));
A339742(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA353471(d), s += A339742(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n squarefree} A339661(n/d).

More terms from Antti Karttunen, May 02 2022

A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A045778 at a(24) = 4, A045778(24) = 5.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into distinct factors > 1.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Sets of multisets are generally not transitive. For example, we have arrows: {{1},{2},{1,2}}: {1,1,2,2} -> {1,2,3} and {{1,2},{3}}: {1,2,3} -> {3,3}, but there is no set of multisets {1,1,2,2} -> {3,3}.

			The prime indices of 24 are {1,1,1,2}, with 5 partitions into a set of multisets:
  {{1,1,1,2}}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{2},{1,1}}
with block-sums: {5}, {1,4}, {2,3}, {2,3}, {1,2,2}, of which 4 are distinct, so a(24) = 4.

Before taking sums we had A045778.
If each block is a set we have A381441, before sums A050326.
For distinct block-sums instead of blocks we have A381637, before sums A321469.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For set systems with distinct sums (A381633) see A381634, zeros A293243.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on sets of multisets: A261049, A317776, A317775, A296118, A318286.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A066328, A213385, A213427, A299200, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&]]],{n,100}]

a(A002110(n)) = A066723(n).

A382201 MM-numbers of sets of sets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 145, 146, 149, 155, 157, 158, 163, 165

Offset: 1

Gus Wiseman, Mar 21 2025

nonn

First differs from A302494 in lacking 143, corresponding to the multiset partition {{1,2},{3}}.
Also products of prime numbers of squarefree index such that the factors all have distinct sums of prime indices.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  22: {{},{3}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  39: {{1},{1,2}}

Set partitions of this type are counted by A275780.
Twice-partitions of this type are counted by A279785.
For just sets of sets we have A302478.
For distinct blocks instead of block-sums we have A302494.
For equal instead of distinct sums we have A302497.
For just distinct sums we have A326535.
For normal multiset partitions see A326519, A326533, A326537, A381718.
Factorizations of this type are counted by A381633. See also A001055, A045778, A050320, A050326, A321455, A321469, A382080.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A005117, A007716, A293511, A302242, A319899, A326534, A368100, A368101, A381635, A382215.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]

Equals A302478 /\ A326535.

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 210, 436, 894

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A382216 at a(9) = 210, A382216(9) = 208.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292432.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
The strong version is A381996, complement A292444.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
For distinct sums we have A382216, complement A382202.
The case of a unique choice is counted by A382458, distinct sums A382459.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A317532, A326517, A326519, A381990, A381992, A382428, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, A382201.

facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d, Rest[Divisors[n]]}]];
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]]]];
Table[Sum[Length[Select[mps[y], UnsameQ@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A162143 Numbers that are the squares of the product of three distinct primes.

Original entry on oeis.org

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684, 119025, 125316, 127449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Numbers that are the product of exactly 3 distinct squares of primes (p^2*q^2*r^2).

			900 = 2^2*3^2*5^2, 1764 = 2^2*3^2*7^2, 4356 = 2^2*3^2*11^2, ..

Cf. A007304, A006881, A050326, A162142, A085548, A085964, A085966.

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A162143List := upto -> seq(n^2, n=select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

fQ[n_]:=Last/@FactorInteger[n]=={2,2,2}; Select[Range[100000], f]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162143(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 29 2024

def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n*n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

a(n) = A007304(n)^2.
A050326(a(n)) = 8. - Reinhard Zumkeller, May 03 2013
Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 = (A085548^3 + 2*A085966 - 3*A085548*A085964)/6 = 0.0036962441..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Edited by N. J. A. Sloane, Jun 27 2009

A212168 Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)).

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Matthew Vandermast, May 22 2012

nonn

A225230(a(n)) > 1; A050326(a(n)) > 1. - Reinhard Zumkeller, May 03 2013

Subsequence of A130092. - Ivan N. Ianakiev, Sep 17 2019

			10 = 2^1*5^1 has 2 distinct prime factors, hence 2 positive exponents in its prime factorization (although the 1s are often left implicit). 2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 belongs to the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages)

Complement of A212165. See also A212164, A212166-A212167.

Subsequence of A188654.

import Data.List (findIndices)
a212168 n = a212168_list !! (n-1)
a212168_list = map (+ 1) $ findIndices (> 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] < Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)
Select[Range[200],Max[FactorInteger[#][[All,2]]]Harvey P. Dale, Nov 21 2018 *)

is(n,f=factor(n))=my(e=f[,2]); #e && vecmax(e)<#e \\ Charles R Greathouse IV, Jan 09 2022

A339661 Number of factorizations of n into distinct squarefree semiprimes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 19 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
                      (10*14*33)  (10*39*77)   (6*14*15*55)
                      (10*21*22)  (14*33*65)   (6*15*22*35)
                      (14*15*22)  (14*39*55)   (10*14*15*33)
                                  (15*22*91)   (10*15*21*22)
                                  (15*26*77)
                                  (21*22*65)
                                  (21*26*55)
                                  (22*35*39)
                                  (26*33*35)
                                  (6*35*143)
                                  (10*21*143)
                                  (14*15*143)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
A050326 allows all squarefree numbers, non-strict case A050320.
A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

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

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

More terms and secondary offset added by Antti Karttunen, May 02 2022

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

Offset: 0

Gus Wiseman, Mar 20 2025

			The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
The a(1) = 1 through a(12) = 13 partitions:
  1  2  3  4    5    6     7    8      9      A      B      C
           211  221  411   322  332    441    433    443    552
                311  2211  331  422    522    442    533    633
                           511  611    711    622    551    822
                                3311   42111  811    722    A11
                                32111         3322   911    4422
                                              4411   42221  5511
                                              32221  53111  33321
                                              43111  62111  52221
                                              52111         54111
                                                            63111
                                                            72111
                                                            3222111

Normal multiset partitions of this type are counted by A116539, see A381718.
These partitions are ranked by A293511.
MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
For distinct block-sums instead of blocks we have A382460, ranked by A381870.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

cp's OEIS Frontend

A382200 Numbers that can be written as a product of distinct squarefree numbers.

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A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

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A381452 Number of multisets that can be obtained by partitioning the prime indices of n into a set of multisets and taking their sums.

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A382201 MM-numbers of sets of sets with distinct sums.

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A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

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A050345 Number of ways to factor n into distinct factors with one level of parentheses.

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A162143 Numbers that are the squares of the product of three distinct primes.

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