A320462 -id:A320462 - cp's OEIS frontend

A320655 Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

The characteristic function of nonzero terms is A065043. - R. J. Mathar, Jan 18 2021

			The a(900) = 5 factorizations into semiprimes:
  900 = (4*9*25)
  900 = (4*15*15)
  900 = (6*6*25)
  900 = (6*10*15)
  900 = (9*10*10)
The a(900) = 5 multiset partitions into pairs:
  {{1,1},{2,2},{3,3}}
  {{1,1},{2,3},{2,3}}
  {{1,2},{1,2},{3,3}}
  {{1,2},{1,3},{2,3}}
  {{2,2},{1,3},{1,3}}

Antti Karttunen, Table of n, a(n) for n = 1..65537

The positions of zeros are A026424.

Cf. A001055, A001222, A001358, A007716, A007717, A056239, A112798, A318871, A318953, A320462, A320656, A320658, A320659.

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

A320655(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A320655(n/d, d))); (s)); \\ Antti Karttunen, Dec 06 2020

Data section extended up to 105 terms by Antti Karttunen, Dec 06 2020

A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

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

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The a(4620) = 6 factorizations into squarefree semiprimes:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(4620) = 6 multiset partitions into strict pairs:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}
The a(69300) = 10 factorizations into squarefree semiprimes:
  69300 = (6*6*35*55)
  69300 = (6*10*15*77)
  69300 = (6*10*21*55)
  69300 = (6*10*33*35)
  69300 = (6*14*15*55)
  69300 = (6*15*22*35)
  69300 = (10*10*21*33)
  69300 = (10*14*15*33)
  69300 = (10*15*21*22)
  69300 = (14*15*15*22)
The a(69300) = 10 multiset partitions into strict pairs:
  {{1,2},{1,2},{3,4},{3,5}}
  {{1,2},{1,3},{2,3},{4,5}}
  {{1,2},{1,3},{2,4},{3,5}}
  {{1,2},{1,3},{2,5},{3,4}}
  {{1,2},{1,4},{2,3},{3,5}}
  {{1,2},{2,3},{1,5},{3,4}}
  {{1,3},{1,3},{2,4},{2,5}}
  {{1,3},{1,4},{2,3},{2,5}}
  {{1,3},{2,3},{2,4},{1,5}}
  {{1,4},{2,3},{2,3},{1,5}}.
The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - _Antti Karttunen_, Nov 02 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.

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

A320656(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&issquarefree(d)&&2==bigomega(d), s += A320656(n/d, d))); (s)); \\ Antti Karttunen, Nov 02 2022

a(A002110(n)) = A123023(n). - Antti Karttunen, Nov 02 2022

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Nov 02 2022

A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117

Offset: 1

Gus Wiseman, Oct 13 2018

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. The n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}

Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

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[100],normQ[primeMS/@primeMS[#]]&]

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

Original entry on oeis.org

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

Original entry on oeis.org

1, 1, 4, 7, 21, 40, 106, 216, 534, 1139, 2715, 5962, 14012, 31420, 73484, 167617, 392714, 908600, 2140429, 5015655, 11905145, 28228533, 67590229, 162067916, 391695348, 949359190, 2316618809, 5673557284, 13979155798, 34583650498, 86034613145, 214948212879

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}
         {{1,2}}    {{1},{2,2}}    {{1,1},{2,2}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
         {{1},{2}}  {{2},{1,2}}    {{1,2},{2,2}}
                    {{1},{1},{1}}  {{1,2},{3,3}}
                    {{1},{2},{2}}  {{1,2},{3,4}}
                    {{1},{2},{3}}  {{1,3},{2,3}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{2,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
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}
gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((1+r)\2)*x^(2*r))}
a(n)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!} \\ Andrew Howroyd, Oct 26 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 26 2018

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691

Offset: 1

Gus Wiseman, Oct 13 2018

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. The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   667: {{2,2},{1,3}}
  1261: {{3,3},{1,2}}
  1363: {{1,3},{2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2093: {{1,1},{2,2},{1,2}}
  2117: {{1,3},{2,4}}
  2639: {{1,1},{1,2},{1,3}}
  4277: {{1,1},{1,2},{2,3}}
  4669: {{1,1},{2,2},{1,3}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 377, 611, 1363, 1937, 2021, 2117, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 143663, 146653, 147533, 153023, 159659, 167243, 170839, 203087, 237679, 243893, 265369, 271049, 276877, 290029, 301129, 315433, 467711

Offset: 1

Gus Wiseman, Oct 13 2018

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. The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1363: {{1,3},{2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
  16211: {{1,2},{1,3},{1,4}}
  17719: {{1,2},{1,3},{2,3}}
  26273: {{1,2},{1,4},{2,3}}
  27521: {{1,2},{1,3},{2,4}}
  44603: {{1,2},{2,3},{2,4}}
  56173: {{1,2},{1,3},{3,4}}
  58609: {{1,3},{1,4},{2,3}}
  83291: {{1,2},{1,4},{3,4}}
  91031: {{1,3},{1,4},{2,4}}
  91039: {{1,2},{2,3},{3,4}}
  99499: {{1,3},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A001222, A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A302491, A305052.

Cf. A320456, A320459, A320461, A320462, A320463.

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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A029862 Expansion of q^(5/24) / (eta(q) * eta(q^2)^2) in powers of q.

Original entry on oeis.org

1, 1, 4, 5, 14, 18, 41, 54, 109, 145, 267, 357, 618, 826, 1359, 1815, 2872, 3824, 5859, 7774, 11600, 15329, 22362, 29425, 42113, 55167, 77648, 101267, 140479, 182395, 249789, 322906, 437199, 562755, 754171, 966713, 1283630, 1638716, 2157763

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there are 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

Also the number of non-isomorphic multiset partitions of weight n using singletons or pairs where no vertex appears more than twice. - Gus Wiseman, Oct 18 2018 (Proved by Andrew Howroyd, Oct 26 2018)

			G.f. = 1 + x + 4*x^2 + 5*x^3 + 14*x^4 + 18*x^5 + 41*x^6 + 54*x^7 + 109*x^8 + ...
G.f. = q^-5 + q^19 + 4*q^43 + 5*q^67 + 14*q^91 + 18*q^115 + 41*q^139 + ...
From _Gus Wiseman_, Oct 27 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions using singletons or pairs where no vertex appears more than twice:
  {{1}}  {{1,1}}    {{1},{2,2}}    {{1,1},{2,2}}      {{1},{2,2},{3,3}}
         {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,3}}      {{1},{2,3},{4,4}}
         {{1},{2}}  {{1},{2},{2}}  {{1,2},{3,4}}      {{1},{2,3},{4,5}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{2,4},{3,4}}
                                   {{1},{1},{2,2}}    {{2},{1,2},{3,3}}
                                   {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{4},{1,2},{3,4}}
                                   {{1},{2},{3,3}}    {{1},{1},{3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{2},{3,3}}
                                   {{1},{3},{2,3}}    {{1},{2},{2},{3,4}}
                                   {{1},{1},{2},{2}}  {{1},{2},{3},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{3},{4,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A001358, A007716, A007717, A037143, A320462, A320655, A320663, A320665.

nmax = 40; CoefficientList[Series[Product[1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = 1/(QP[q]*QP[q^2]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A) * eta(x^2 + A)^2), n))};

Euler transform of period 2 sequence [ 1, 3, ...].
G.f.: Product_{k>0} 1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))). - Michael Somos, Mar 23 2003
a(n) ~ exp(2*Pi*sqrt(n/3))/(6*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Sep 07 2015

A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 169, 377, 611, 1363, 1937, 2021, 2117, 2197, 4901, 7943, 10933, 16211, 17719, 25181, 26273, 27521, 28561, 28717, 39527, 44603, 56173, 58609, 61393, 63713, 64061, 83291, 86903, 91031, 91039, 94987, 99499, 103259, 141401, 142129, 143663, 146653, 147533

Offset: 1

Gus Wiseman, Oct 13 2018

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. The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    169: {{1,2},{1,2}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1363: {{1,3},{2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   2197: {{1,2},{1,2},{1,2}}
   4901: {{1,2},{1,2},{1,3}}
   7943: {{1,2},{1,2},{2,3}}
  10933: {{1,2},{1,3},{1,3}}
  16211: {{1,2},{1,3},{1,4}}
  17719: {{1,2},{1,3},{2,3}}
  25181: {{1,2},{1,2},{3,4}}
  26273: {{1,2},{1,4},{2,3}}
  27521: {{1,2},{1,3},{2,4}}
  28561: {{1,2},{1,2},{1,2},{1,2}}
  28717: {{1,2},{2,3},{2,3}}
  39527: {{1,3},{1,3},{2,3}}
  44603: {{1,2},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052, A305078.

Cf. A320456, A320458, A320461, A320462, A320464.

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[100000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 19, 37, 49, 53, 61, 89, 91, 113, 131, 133, 151, 161, 169, 223, 247, 251, 259, 281, 299, 311, 329, 343, 359, 361, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 637, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 931, 953

Offset: 1

Gus Wiseman, Oct 14 2018

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. The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
    1: {}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  133: {{1,1},{1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  259: {{1,1},{1,1,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}
  311: {{1,1,1,1,1,1}}
  329: {{1,1},{2,3}}
  343: {{1,1},{1,1},{1,1}}

Wikipedia, Hypergraph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320462, A320463, A320464, A320532.

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[normQ[primeMS/@primeMS[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

cp's OEIS Frontend

A320655 Number of factorizations of n into semiprimes. Number of multiset partitions of the multiset of prime factors of n, into pairs.

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A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

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A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

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A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

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A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

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A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

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A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

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A029862 Expansion of q^(5/24) / (eta(q) * eta(q^2)^2) in powers of q.

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