OFFSET
1,2
COMMENTS
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}}.
EXAMPLE
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}}
MATHEMATICA
sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !sqfsemiQ[PrimePi[p]]]&]
CROSSREFS
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
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 12 2021
STATUS
approved