A050320 -id:A050320 - cp's OEIS frontend

A050325 Number of factorizations into squarefree factors indexed by prime signatures: A050320(A025487).

1, 1, 1, 2, 1, 2, 1, 2, 5, 1, 3, 2, 6, 1, 3, 2, 6, 1, 3, 9, 2, 15, 4, 6, 1, 3, 10, 2, 21, 4, 6, 1, 3, 10, 2, 22, 4, 16, 6, 1, 13, 3, 35, 5, 10, 2, 22, 4, 19, 6, 1, 14, 3, 52, 41, 5, 10, 2, 22, 4, 20, 6, 1, 14, 3, 83, 42, 5, 26, 10, 2, 66, 17, 22, 4, 20, 55, 6, 6, 1, 14, 3, 93

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3300

Cf. A025487, A050320.

read(transforms):
a050320 := BFILETOLIST("b050320.txt") :
for n from 1 do
    a050325 := a050320[A025487(n)] ;
    printf("%d %d\n",n,a050325) ;
end do: # R. J. Mathar, May 26 2017

a050320 = Cases[Import["https://oeis.org/A050320/b050320.txt", "Table"], {, }][[All, 2]];
lpe = {}; a025487 = {1};
Do[pe = Sort[FactorInteger[n][[All, 2]]]; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[a025487, n]], {n, 2, 10^4}];
A050325[n_] := a050320[[a025487[[n]]]];
Array[A050325, 83] (* Jean-François Alcover, Apr 06 2020 *)

a(n) = A050320(A025487(n)).

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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, 5, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 5, 1, 1, 2, 5, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2, 0, 1, 4, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 5, 1

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

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).
a(A212164(n)) = 0; a(A212166(n)) = 1; a(A006881(n)) = 2; a(A190107(n)) = 3; a(A085987(n)) = 4; a(A225228(n)) = 5; a(A179670(n)) = 7; a(A162143(n)) = 8; a(A190108(n)) = 11; a(A212167(n)) > 0; a(A212168(n)) > 1. - Reinhard Zumkeller, May 03 2013
The comment that a(A212164(n)) = 0 is incorrect. For example, 3600 belongs to A212164 but a(3600) = 1. The positions of zeros in this sequence are A293243. - Gus Wiseman, Oct 10 2017

			The a(30) = 5 factorizations are: 2*3*5, 2*15, 3*10, 5*6, 30. The a(180) = 5 factorizations are: 2*3*5*6, 2*3*30, 2*6*15, 3*6*10, 6*30. - _Gus Wiseman_, Oct 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.

import Data.List (subsequences, genericIndex)
a050326 n = genericIndex a050326_list (n-1)
a050326_list = 1 : f 2 where
   f x = (if x /= s then a050326 s
                    else length $ filter (== x) $ map product $
                         subsequences $ tail $ a206778_row x) : f (x + 1)
         where s = a046523 x
-- Reinhard Zumkeller, May 03 2013

N:= 1000: # to get a(1)..a(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:
convert(A,list); # Robert Israel, Oct 10 2017

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Oct 10 2017 *)

Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).

a(n) = A050327(A101296(n)). - R. J. Mathar, May 26 2017

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 1, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 6, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 3, 1, 0, 6, 1, 1, 1, 4, 0, 6, 1, 2, 1, 1, 1, 10, 0, 2, 2, 5, 0, 3, 0, 4, 3

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, g(n$2, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

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],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 5, 3, 2, 1, 6, 1, 2, 3, 15, 1, 9, 1, 6, 3, 2, 1, 21, 4, 2, 16, 6, 1, 10, 1, 52, 3, 2, 4, 35, 1, 2, 3, 22, 1, 10, 1, 6, 19, 2, 1, 83, 5, 13, 3, 6, 1, 66, 4, 22, 3, 2, 1, 41, 1, 2, 20, 203, 4, 10, 1, 6, 3, 14, 1, 153, 1, 2, 26, 6, 5, 10, 1

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(12) = 6 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A001055, A007716, A049311, A116540, A181821, A188392, A255906.

Cf. A318283, A318284, A318286, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,80}]

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1],s[2])+1), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A050320(A181821(n)).
From Andrew Howroyd, Dec 10 2018:(Start)
a(p) = 1 for prime(p).
a(prime(i)*prime(j)) = min(i,j) + 1.
a(prime(n)^k) = A188392(n,k). (End)

A302478 Products of prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 66, 67, 68, 72, 73, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 90, 93, 94

Offset: 1

Gus Wiseman, Apr 08 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
01:  {}
02:  {{}}
03:  {{1}}
04:  {{},{}}
05:  {{2}}
06:  {{},{1}}
08:  {{},{},{}}
09:  {{1},{1}}
10:  {{},{2}}
11:  {{3}}
12:  {{},{},{1}}
13:  {{1,2}}
15:  {{1},{2}}
16:  {{},{},{},{}}
17:  {{4}}
18:  {{},{1},{1}}
20:  {{},{},{2}}
22:  {{},{3}}
24:  {{},{},{},{1}}
25:  {{2},{2}}
26:  {{},{1,2}}
27:  {{1},{1},{1}}
29:  {{1,3}}
30:  {{},{1},{2}}
31:  {{5}}
32:  {{},{},{},{},{}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001222, A003963, A005117, A007716, A050320, A056239, A063834, A076610, A270995, A275024, A281113, A296119, A301753, A302242, A302243, A302491.

Select[Range[100],Or[#===1,And@@SquareFreeQ/@PrimePi/@FactorInteger[#][[All,1]]]&]

ok(n)={!#select(p->!issquarefree(primepi(p)), factor(n)[,1])} \\ Andrew Howroyd, Aug 26 2018

A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 27 2009

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

			Triangle begins:
   1....... 1;
   2....... 1;
   3....... 1;
   4..... 1,2;
   5....... 1;
   6..... 1,2;
   7....... 1;
   8..... 1,2;
   9..... 1,3;
  10..... 1,2;
  11....... 1;
  12... 1,2,3;
  13....... 1;
  14..... 1,2;
  15..... 1,3;
  16... 1,2,4;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Initial terms are A000012.
Final terms are A033676.
Row lengths are A038548 (number of inferior divisors).
Row sums are A066839 (sum of inferior divisors).
The prime terms are counted by A063962.
The odd terms are counted by A069288.
Row products are A072499.
Row LCMs are A072504.
The superior version is A161908.
The squarefree terms are counted by A333749.
The prime-power terms are counted by A333750.
The strictly superior version is A341673.
The strictly inferior version is A341674.
A001221 counts prime divisors, with sum A001414.
A000005 counts divisors, listed by A027750 with sum A000203.
A056924 count strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A217581.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000196, A001055, A001248, A006530, A020639, A050320, A068101, A161901.

a161906 n k = a161906_tabf !! (n-1) !! (k-1)
a161906_row n = a161906_tabf !! (n-1)
a161906_tabf = zipWith (\m ds -> takeWhile ((<= m) . (^ 2)) ds)
                       [1..] a027750_tabf'
-- Reinhard Zumkeller, Jun 24 2015, Mar 08 2013

div[n_] := Select[Divisors[n], # <= Sqrt[n] &]; div /@ Range[48] // Flatten (* Amiram Eldar, Nov 13 2020 *)

row(n) = select(x->(x<=sqrt(n)), divisors(n)); \\ Michel Marcus, Nov 13 2020

More terms from Sean A. Irvine, Nov 29 2010

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 3, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 3, 3, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 3, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
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 ranked 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).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
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.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]]]],{n,100}]

a(A002110(n)) = A381808(n).

A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

12, 60, 63, 84, 120, 126, 132, 156, 204, 228, 252, 276, 300, 315, 325, 348, 372, 420, 444, 492, 504, 516, 560, 564, 588, 630, 636, 650, 660, 693, 708, 720, 732, 780, 804, 819, 840, 852, 876, 924, 931, 948, 975, 996, 1008, 1020, 1068, 1071, 1092, 1140, 1164

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

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.
Also numbers that cannot be written as a product of prime powers > 1 with distinct sums of prime indices (A056239).
Contains no squarefree numbers.
Conjecture: These are the zeros of A382876.

			The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
  {{2},{1,1},{3,3}}
  {{1},{1},{2},{3,3}}
  {{2},{3},{3},{1,1}}
  {{1},{1},{2},{3},{3}}
but none of these has distinct block-sums, so 300 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  300: {1,1,2,3,3}

More on multiset partitions into constant blocks: A006171, A279784, A295935.
These are the positions of 0 in A381635, after taking block-sums A381716.
Partitions of this type are counted by A381717.
For strict instead of constant blocks we have A381806, zeros of A381633.
For equal instead of distinct block-sums we have A381871.
A000688 counts multiset partitions into constant, see A381455 (upper), A381453 (lower).
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A050361 counts multiset partitions into distinct constant blocks, after sums A381715.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A005117, A050320, A059404, A213242, A293243, A299202, A300385, A381078, A381454, A381634.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]

cp's OEIS Frontend

A050325 Number of factorizations into squarefree factors indexed by prime signatures: A050320(A025487).

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A381432 Heinz numbers of section-sum partitions. Union of A381431.

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A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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A339846 Number of even-length factorizations of n into factors > 1.

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A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

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A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

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A302478 Products of prime numbers of squarefree index.

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A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

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