A316476 -id:A316476 - cp's OEIS frontend

A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

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

Offset: 1

Gus Wiseman, Sep 15 2019

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. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476.

			The a(26) = 4 factorizations of 45 into stable numbers:
  (3*3*5)
  (3*15)
  (5*9)
  (45)
The a(201) = 11 multiset partitions of the prime indices of 495 into stable multisets:
  {{2},{2},{3},{5}}
  {{2},{2},{3,5}}
  {{2},{3},{2,5}}
  {{2},{5},{2,3}}
  {{2},{2,3,5}}
  {{3},{2,2},{5}}
  {{3},{2,2,5}}
  {{2,2},{3,5}}
  {{5},{2,2,3}}
  {{2,3},{2,5}}
  {{2,2,3,5}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A001055, A303362, A305149.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
y=Select[Range[nn],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

A326704 BII-numbers of antichains of nonempty sets.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524

Offset: 1

Gus Wiseman, Jul 21 2019

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.

			The sequence of all antichains of nonempty sets together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..7580
John Tyler Rascoe, Python Program.

Antichains of sets are counted by A000372.
Antichains of nonempty sets are counted by A014466.
MM-numbers of antichains of multisets are A316476.
BII-numbers of chains of nonempty sets are A326703.
Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],stableQ[bpe/@bpe[#],SubsetQ]&]

Python
```
# see linked program
```

A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

			20 is the Matula-Goebel number of the tree (oo((o))), which has 4 distinct terminal subtrees: {(oo((o))), ((o)), (o), o}. So a(20) = 4.
See also illustrations in A061773.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Matula-Göbel numbers

One more than A324923.

Cf. A000081, A007097, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476.

ids[n_]:=Union@@FixedPointList[Union@@(Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]&/@#)&,{n}];
Table[Length[ids[n]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023
A317713(n) = (1+A324923(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = 1+A324923(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 15, 2, 17, 10, 23, 2, 39, 2, 46, 18, 58, 2, 95, 8, 103, 31, 139, 2, 219, 3, 232, 59, 299, 22, 452, 4, 492, 104, 645, 5, 920, 5, 1006, 204, 1258, 8, 1785, 21, 1994, 302, 2442, 11, 3366, 71, 3738, 497, 4570, 18, 6253, 24, 6849

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

A partition with no two distinct parts relatively prime is said to be intersecting.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        63         55
              1111         42               62        333        64
                           222              422       111111111  82
                           111111           2222                 442
                                            11111111             622
                                                                 4222
                                                                 22222
                                                                 1111111111

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

The Heinz numbers of these partitions are A328867 (strict case is A318719).
The relatively prime case is A328672.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
The version involving all parts (not just distinct ones) is A200976.
Cf. A000837, A202425, A305148, A305854, A306006, A316476, A326910.

Table[Length[Select[IntegerPartitions[n],And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]&]],{n,0,20}]

a(n > 0) = A200976(n) + 1.

A200976 Number of partitions of n such that each pair of parts (if any) has a common factor.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 1, 5, 3, 8, 1, 14, 1, 16, 9, 22, 1, 38, 1, 45, 17, 57, 1, 94, 7, 102, 30, 138, 1, 218, 2, 231, 58, 298, 21, 451, 3, 491, 103, 644, 4, 919, 4, 1005, 203, 1257, 7, 1784, 20, 1993, 301, 2441, 10, 3365, 70, 3737, 496, 4569, 17, 6252, 23, 6848

Offset: 0

Alois P. Heinz, Nov 29 2011

nonn

a(n) is different from A018783(n) for n = 0, 31, 37, 41, 43, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, ... .

Every pair of (possibly equal) parts has a common factor > 1. These partitions are said to be (pairwise) intersecting. - Gus Wiseman, Nov 04 2019

			a(0) = 1: [];
a(4) = 2: [2,2], [4];
a(9) = 3: [3,3,3], [3,6], [9];
a(31) = 2: [6,10,15], [31];
a(41) = 4: [6,10,10,15], [6,15,20], [6,14,21], [41].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350 (terms 0..250 from Alois P. Heinz)
L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8

Cf. A018783.
The version with only distinct parts compared is A328673.
The relatively prime case is A202425.
The strict case is A318717.
The version for non-isomorphic multiset partitions is A319752.
The version for set-systems is A305843.
Cf. A000837, A305148, A305854, A306006, A316476, A328672, A328867, A328868.

b:= proc(n, j, s) local ok, i;
      if n=0 then 1
    elif j<2 then 0
    else ok:= true;
         for i in s while ok do ok:= evalb(igcd(i, j)<>1) od;
         `if`(ok, add(b(n-j*k, j-1, [s[], j]), k=1..n/j), 0) +b(n, j-1, s)
      fi
    end:
a:= n-> b(n, n, []):
seq(a(n), n=0..62);

b[n_, j_, s_] := Module[{ok, i, is}, Which[n == 0, 1, j < 2, 0, True, ok = True; For[is = 1, is <= Length[s] && ok, is++, i = s[[is]]; ok = GCD[i, j] != 1]; If[ok, Sum[b[n-j*k, j-1, Append[s, j]], {k, 1, n/j}], 0] + b[n, j-1, s]]]; a[n_] := b[n, n, {}]; Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Dec 26 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],And[And@@(GCD[##]>1&)@@@Select[Tuples[Union[#],2],LessEqual@@#&]]&]],{n,0,20}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328673(n) - 1. - Gus Wiseman, Nov 04 2019

A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227

Offset: 1

Gus Wiseman, Nov 18 2019

nonn

A weak antichain of multisets is a multiset of multisets, none of which is a proper subset of any other.

			The sequence of terms tother with their corresponding clutters begins:
   1: {}              37: {{1,1,2}}            91: {{1,1},{1,2}}
   2: {{}}            41: {{6}}                97: {{3,3}}
   3: {{1}}           43: {{1,4}}             101: {{1,6}}
   5: {{2}}           47: {{2,3}}             103: {{2,2,2}}
   7: {{1,1}}         49: {{1,1},{1,1}}       107: {{1,1,4}}
   9: {{1},{1}}       53: {{1,1,1,1}}         109: {{10}}
  11: {{3}}           59: {{7}}               113: {{1,2,3}}
  13: {{1,2}}         61: {{1,2,2}}           121: {{3},{3}}
  17: {{4}}           67: {{8}}               125: {{2},{2},{2}}
  19: {{1,1,1}}       71: {{1,1,3}}           127: {{11}}
  23: {{2,2}}         73: {{2,4}}             131: {{1,1,1,1,1}}
  25: {{2},{2}}       79: {{1,5}}             137: {{2,5}}
  27: {{1},{1},{1}}   81: {{1},{1},{1},{1}}   139: {{1,7}}
  29: {{1,3}}         83: {{9}}               149: {{3,4}}
  31: {{5}}           89: {{1,1,1,2}}         151: {{1,1,2,2}}

Connected numbers are A305078.
Stable numbers are A316476.
Clutters (of sets) are A048143.
Cf. A056239, A112798, A289509, A302242, A302494, A304716, A318991, A319837, A320275, A320456, A328514, A329553, A329555.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],And[stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]<=1]&]

Equals {1} followed by the intersection of A305078 and A316476.

A339113 Products of primes of squarefree semiprime index (A322551).

Original entry on oeis.org

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

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

Also MM-numbers of labeled multigraphs (without uncovered vertices). 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. 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 sequence of terms together with the corresponding multigraphs begins:
      1: {}               233: {{2,7}}          487: {{2,11}}
     13: {{1,2}}          257: {{3,5}}          491: {{1,15}}
     29: {{1,3}}          269: {{2,8}}          499: {{3,8}}
     43: {{1,4}}          271: {{1,10}}         559: {{1,2},{1,4}}
     47: {{2,3}}          293: {{1,11}}         577: {{1,16}}
     73: {{2,4}}          313: {{3,6}}          607: {{2,12}}
     79: {{1,5}}          347: {{2,9}}          611: {{1,2},{2,3}}
    101: {{1,6}}          373: {{1,12}}         631: {{3,9}}
    137: {{2,5}}          377: {{1,2},{1,3}}    647: {{1,17}}
    139: {{1,7}}          389: {{4,5}}          653: {{4,7}}
    149: {{3,4}}          421: {{1,13}}         673: {{1,18}}
    163: {{1,8}}          439: {{3,7}}          677: {{2,13}}
    167: {{2,6}}          443: {{1,14}}         727: {{2,14}}
    169: {{1,2},{1,2}}    449: {{2,10}}         751: {{4,8}}
    199: {{1,9}}          467: {{4,6}}          757: {{1,19}}

These primes (of squarefree semiprime index) are listed by A322551.
The strict (squarefree) case is A309356.
The prime instead of squarefree semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of squarefree semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The semiprime instead of squarefree semiprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices, which are listed by A112798.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339561 lists products of distinct squarefree semiprimes (ranking: A339560).
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A289509, A320461.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336

Offset: 0

Gus Wiseman, Mar 18 2024

nonn

Also strict integer partitions such that the number of parts is equal to the number of distinct divisors of all parts.

			The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
  531  721   731   B1    751   D1    B31    D21    B51    H1     B71
       4321  5321  5421  931   B21   7521   7531   D31    9531   D51
                   6321  7321  7421  8421   64321  B321   A521   B521
                                     9321          65321  B421   D321
                                     54321         74321  75321  75421
                                                          84321  76321
                                                                 94321

The LHS is represented by A001221, distinct case of A001222.
The RHS is represented by A370820, for prime factors A303975.
Strict case of A371130 (ranks A370802) and A371178 (ranks A371177).
The complement is counted by A371180, non-strict A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A319055, A370803, A370808, A371171, A371172, A371173.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]

A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 128, 129, 131, 133, 137, 139, 147, 149

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

A partition with no two distinct parts relatively prime is said to be intersecting.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328673.
The strict case is A318719.
The relatively prime version is A328868.
A ranking using binary indices is A326910.
The version for non-isomorphic multiset partitions is A319752.
The version for divisibility (instead of relative primality) is A316476.
Cf. A000837, A056239, A112798, A200976, A289509, A303283, A305843, A318715, A318716, A328336.

Select[Range[100],And@@(GCD[##]>1&)@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (without uncovered vertices). 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. 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 sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

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

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

cp's OEIS Frontend

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A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

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A328673 Number of integer partitions of n in which no two distinct parts are relatively prime.

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A200976 Number of partitions of n such that each pair of parts (if any) has a common factor.

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A329559 MM-numbers of multiset clutters (connected weak antichains of multisets).

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A339113 Products of primes of squarefree semiprime index (A322551).

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A371128 Number of strict integer partitions of n containing all distinct divisors of all parts.

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