OFFSET
1,2
COMMENTS
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}}.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
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)
MAPLE
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
MATHEMATICA
semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !semiQ[PrimePi[p]]]&]
CROSSREFS
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
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
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.
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Mar 12 2021
STATUS
approved