A339113 -id:A339113 - cp's OEIS frontend

A339620 Heinz numbers of non-multigraphical partitions of even numbers.

3, 7, 10, 13, 19, 21, 22, 28, 29, 34, 37, 39, 43, 46, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 155, 156, 159, 163, 166, 171, 172, 173

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);
(2) n can be factored into squarefree semiprimes;
(3) the unordered prime signature of n is multigraphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         53: {16}          94: {1,15}
      7: {4}         55: {3,5}        101: {26}
     10: {1,3}       57: {2,8}        102: {1,2,7}
     13: {6}         61: {18}         107: {28}
     19: {8}         62: {1,11}       111: {2,12}
     21: {2,4}       66: {1,2,5}      113: {30}
     22: {1,5}       71: {20}         115: {3,9}
     28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
     29: {10}        79: {22}         117: {2,2,6}
     34: {1,7}       82: {1,13}       118: {1,17}
     37: {12}        85: {3,7}        129: {2,14}
     39: {2,6}       87: {2,10}       130: {1,3,6}
     43: {14}        88: {1,1,1,5}    131: {32}
     46: {1,9}       89: {24}         133: {4,8}
     52: {1,1,6}     91: {4,6}        134: {1,19}
For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a multigraph (they have loops), the Heinz number 66 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A000070 counts these partitions.
A300061 is a superset.
A320891 has image under A181819 equal to this set of terms.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620 [this sequence]).
- A209816 counts multigraphical partitions (A320924).
- A147878 counts connected multigraphical partitions (A320925).
- 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:
- A027187 has no additional conditions (A028260).
- 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. A001055, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.

prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&prpts[nrmptn[#]]=={}&]

Equals A300061 \ A320924.

For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.

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]]]&]

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

A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 39, 41, 43, 47, 51, 55, 59, 65, 67, 73, 79, 83, 85, 87, 93, 101, 109, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237, 241, 249

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal 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}}.
Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime (A006881).

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}              55: {{2},{3}}      137: {{2,5}}
     3: {{1}}           59: {{7}}          139: {{1,7}}
     5: {{2}}           65: {{2},{1,2}}    141: {{1},{2,3}}
    11: {{3}}           67: {{8}}          143: {{3},{1,2}}
    13: {{1,2}}         73: {{2,4}}        145: {{2},{1,3}}
    15: {{1},{2}}       79: {{1,5}}        149: {{3,4}}
    17: {{4}}           83: {{9}}          155: {{2},{5}}
    29: {{1,3}}         85: {{2},{4}}      157: {{12}}
    31: {{5}}           87: {{1},{1,3}}    163: {{1,8}}
    33: {{1},{3}}       93: {{1},{5}}      165: {{1},{2},{3}}
    39: {{1},{1,2}}    101: {{1,6}}        167: {{2,6}}
    41: {{6}}          109: {{10}}         177: {{1},{7}}
    43: {{1,4}}        123: {{1},{6}}      179: {{13}}
    47: {{2,3}}        127: {{11}}         187: {{3},{4}}
    51: {{1},{4}}      129: {{1},{1,4}}    191: {{14}}

The version with full loops covering an initial interval is A320461.
The case covering an initial interval is A340018.
The version with full loops is A340020.
A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A322551 lists primes of squarefree semiprime index.
A330944 counts nonprime prime indices.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 553, 559, 577, 607, 611, 631, 647, 653, 661, 667, 673, 677

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.
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}}.
Also products of distinct primes whose prime indices are semiprimes, where a semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
      1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
      7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
     13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
     23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
     29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
     43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
     47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
     73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
     79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
     91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
     97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
    101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
    137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
    139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
    149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}

The case with only one edge is A106349.
The case covering an initial interval is A320461.
The version allowing multiple edges is A339112.
The half-loop version covering an initial interval is A340018.
The half-loop version is A340019.
A006450 lists primes of prime index.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

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

A338903 Number of integer partitions of the n-th squarefree semiprime into squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 12, 14, 19, 22, 27, 36, 38, 51, 77, 86, 128, 141, 163, 163, 207, 233, 259, 260, 514, 657, 813, 983, 1010, 1215, 1255, 1720, 2112, 2256, 3171, 3370, 3499, 3864, 4103, 6292, 7313, 7620, 8374, 10650, 17579, 18462, 23034, 25180

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

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

			The a(n) partitions for n = 1, 5, 7, 9, 10, 11, 13:
  6  21    26       34          35        38           46
     15,6  14,6,6   22,6,6      21,14     26,6,6       34,6,6
           10,10,6  14,14,6     15,14,6   22,10,6      26,14,6
                    14,10,10    15,10,10  14,14,10     21,15,10
                    10,6,6,6,6            14,6,6,6,6   22,14,10
                                          10,10,6,6,6  26,10,10
                                                       15,15,10,6
                                                       22,6,6,6,6
                                                       14,14,6,6,6
                                                       14,10,10,6,6
                                                       10,10,10,10,6
                                                       10,6,6,6,6,6,6

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of squarefree semiprimes.
A101048 counts partitions into semiprimes.
A338902 is the not necessarily squarefree version.
A339113 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes.
Cf. A000041, A000607, A065516, A115392, A128301, A320655, A320732, A320892, A320912, A338915, A338916.

nn=100;
sqs=Select[Range[nn],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Length[IntegerPartitions[n,All,sqs]],{n,sqs}]

a(n) = A002100(A006881(n)).

A339740 Non-products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.

A013929 allows only primes.
A320894 does not allow primes (but omega is assumed even).
A339741 is the complement.
A339742 has zeros at these positions.
A339840 allows squares of primes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339617 counts non-graphical partitions of 2n (A339618).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 counts all of even length (A028260).
- A096373/A339737 cannot be partitioned into strict pairs (A320891).
- A338915/A339662 cannot be partitioned into distinct pairs (A320892).
- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

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

A338902 Number of integer partitions of the n-th semiprime into semiprimes.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

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

			The a(1) = 1 through a(33) = 17 partitions of 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, where A-Z = 10-35:
  4  6  9  A   E    F   L     M      P      Q       X
           64  A4   96  F6    994    FA     M4      EA9
               644      966   A66    L4     AA6     F99
                        9444  E44    A96    E66     FE4
                              6664   F64    9944    L66
                              A444   9664   A664    P44
                              64444  94444  E444    9996
                                            66644   AA94
                                            A4444   E964
                                            644444  F666
                                                    FA44
                                                    L444
                                                    96666
                                                    A9644
                                                    F6444
                                                    966444
                                                    9444444

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of semiprimes.
A101048 counts partitions into semiprimes.
A338903 is the squarefree version.
A339112 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A037143 lists primes and semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
Cf. A000041, A000607, A006881, A065516, A115392, A128301, A320656, A320732, A320892, A320912, A338915, A338916, A339113.

nn=100;Table[Length[IntegerPartitions[n,All,Select[Range[nn],PrimeOmega[#]==2&]]],{n,Select[Range[nn],PrimeOmega[#]==2&]}]

a(n) = A101048(A001358(n)).

A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

Original entry on oeis.org

1, 3, 13, 15, 39, 65, 141, 143, 145, 165, 195, 377, 429, 435, 611, 705, 715, 1131, 1363, 1551, 1595, 1833, 1885, 1937, 2021, 2117, 2145, 2235, 2365, 2397, 2409, 2431, 2465, 2805, 3055, 4089, 4147, 4785, 5655, 5811, 6063, 6149, 6235, 6351, 6409, 6721, 6815

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal 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}}.
Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}
     3: {{1}}
    13: {{1,2}}
    15: {{1},{2}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
   141: {{1},{2,3}}
   143: {{3},{1,2}}
   145: {{2},{1,3}}
   165: {{1},{2},{3}}
   195: {{1},{2},{1,2}}
   377: {{1,2},{1,3}}
   429: {{1},{3},{1,2}}
   435: {{1},{2},{1,3}}
   611: {{1,2},{2,3}}
   705: {{1},{2},{2,3}}
   715: {{2},{3},{1,2}}
  1131: {{1},{1,2},{1,3}}

The version with full loops is A320461.
The version not necessarily covering an initial interval is A340019.
MM-numbers of graphs with loops are A340020.
A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
A302242 is the weight of the multiset of multisets with MM-number n.
A302494 lists MM-numbers of sets of sets, with connected case A328514.
A309356 lists MM-numbers of simple graphs.
A322551 lists primes of squarefree semiprime index.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

A371453 Numbers whose binary indices are all squarefree semiprimes.

Original entry on oeis.org

32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184

Offset: 1

Gus Wiseman, Apr 02 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their binary expansions and binary indices begin:
       32:                 100000 ~ {6}
      512:             1000000000 ~ {10}
      544:             1000100000 ~ {6,10}
     8192:         10000000000000 ~ {14}
     8224:         10000000100000 ~ {6,14}
     8704:         10001000000000 ~ {10,14}
     8736:         10001000100000 ~ {6,10,14}
    16384:        100000000000000 ~ {15}
    16416:        100000000100000 ~ {6,15}
    16896:        100001000000000 ~ {10,15}
    16928:        100001000100000 ~ {6,10,15}
    24576:        110000000000000 ~ {14,15}
    24608:        110000000100000 ~ {6,14,15}
    25088:        110001000000000 ~ {10,14,15}
    25120:        110001000100000 ~ {6,10,14,15}
  1048576:  100000000000000000000 ~ {21}

Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
For prime indices instead of binary indices we have A339113, A339112.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A001358 lists squarefree semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.

M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R,list)); # Robert Israel, Apr 04 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000],And@@sqfsemi/@bix[#]&]

def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')

from math import isqrt
from sympy import primepi, primerange
def A371453(n):
    def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A006881(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

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