%I #14 Nov 04 2024 01:39:57

%S 1,7,13,23,29,43,47,49,73,79,91,97,101,137,139,149,161,163,167,169,

%T 199,203,227,233,257,269,271,293,299,301,313,329,343,347,373,377,389,

%U 421,439,443,449,467,487,491,499,511,529,553,559,577,607,611,631,637,647

%N Products of primes of semiprime index (A106349).

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

%C 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}}.

%H Robert Israel, <a href="/A339112/b339112.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):

%e 1: 149: (34) 313: (36)

%e 7: (11) 161: (11)(22) 329: (11)(23)

%e 13: (12) 163: (18) 343: (11)(11)(11)

%e 23: (22) 167: (26) 347: (29)

%e 29: (13) 169: (12)(12) 373: (1C)

%e 43: (14) 199: (19) 377: (12)(13)

%e 47: (23) 203: (11)(13) 389: (45)

%e 49: (11)(11) 227: (44) 421: (1D)

%e 73: (24) 233: (27) 439: (37)

%e 79: (15) 257: (35) 443: (1E)

%e 91: (11)(12) 269: (28) 449: (2A)

%e 97: (33) 271: (1A) 467: (46)

%e 101: (16) 293: (1B) 487: (2B)

%e 137: (25) 299: (12)(22) 491: (1F)

%e 139: (17) 301: (11)(14) 499: (38)

%p N:= 1000: # for terms up to N

%p SP:= {}: p:= 1:

%p for i from 1 do

%p p:= nextprime(p);

%p if 2*p > N then break fi;

%p Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));

%p SP:= SP union Q;

%p od:

%p SP:= sort(convert(SP,list)):

%p PSP:= map(ithprime,SP):

%p R:= {1}:

%p for p in PSP do

%p Rp:= {}:

%p for k from 1 while p^k <= N do

%p Rpk:= select(`<=`,R, N/p^k);

%p Rp:= Rp union map(`*`,Rpk, p^k);

%p od;

%p R:= R union Rp;

%p od:

%p sort(convert(R,list)); # _Robert Israel_, Nov 03 2024

%t semiQ[n_]:=PrimeOmega[n]==2;

%t Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

%Y These primes (of semiprime index) are listed by A106349.

%Y The strict (squarefree) case is A340020.

%Y The prime instead of semiprime version:

%Y primes: A006450

%Y products: A076610

%Y strict: A302590

%Y The nonprime instead of semiprime version:

%Y primes: A007821

%Y products: A320628

%Y odd: A320629

%Y strict: A340104

%Y odd strict: A340105

%Y The squarefree semiprime instead of semiprime version:

%Y strict: A309356

%Y primes: A322551

%Y products: A339113

%Y A001358 lists semiprimes, with odd and even terms A046315 and A100484.

%Y A006881 lists squarefree semiprimes.

%Y A037143 lists primes and semiprimes (and 1).

%Y A056239 gives the sum of prime indices, which are listed by A112798.

%Y A084126 and A084127 give the prime factors of semiprimes.

%Y A101048 counts partitions into semiprimes.

%Y A302242 is the weight of the multiset of multisets with MM-number n.

%Y A305079 is the number of connected components for MM-number n.

%Y A320892 lists even-omega non-products of distinct semiprimes.

%Y A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).

%Y A320912 lists products of distinct semiprimes (Heinz numbers of A338916).

%Y A338898, A338912, and A338913 give the prime indices of semiprimes.

%Y 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).

%Y Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 12 2021