This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339112 #14 Nov 04 2024 01:39:57
%S A339112 1,7,13,23,29,43,47,49,73,79,91,97,101,137,139,149,161,163,167,169,
%T A339112 199,203,227,233,257,269,271,293,299,301,313,329,343,347,373,377,389,
%U A339112 421,439,443,449,467,487,491,499,511,529,553,559,577,607,611,631,637,647
%N A339112 Products of primes of semiprime index (A106349).
%C A339112 A semiprime (A001358) is a product of any two prime numbers.
%C A339112 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 A339112 Robert Israel, <a href="/A339112/b339112.txt">Table of n, a(n) for n = 1..10000</a>
%e A339112 The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
%e A339112 1: 149: (34) 313: (36)
%e A339112 7: (11) 161: (11)(22) 329: (11)(23)
%e A339112 13: (12) 163: (18) 343: (11)(11)(11)
%e A339112 23: (22) 167: (26) 347: (29)
%e A339112 29: (13) 169: (12)(12) 373: (1C)
%e A339112 43: (14) 199: (19) 377: (12)(13)
%e A339112 47: (23) 203: (11)(13) 389: (45)
%e A339112 49: (11)(11) 227: (44) 421: (1D)
%e A339112 73: (24) 233: (27) 439: (37)
%e A339112 79: (15) 257: (35) 443: (1E)
%e A339112 91: (11)(12) 269: (28) 449: (2A)
%e A339112 97: (33) 271: (1A) 467: (46)
%e A339112 101: (16) 293: (1B) 487: (2B)
%e A339112 137: (25) 299: (12)(22) 491: (1F)
%e A339112 139: (17) 301: (11)(14) 499: (38)
%p A339112 N:= 1000: # for terms up to N
%p A339112 SP:= {}: p:= 1:
%p A339112 for i from 1 do
%p A339112 p:= nextprime(p);
%p A339112 if 2*p > N then break fi;
%p A339112 Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
%p A339112 SP:= SP union Q;
%p A339112 od:
%p A339112 SP:= sort(convert(SP,list)):
%p A339112 PSP:= map(ithprime,SP):
%p A339112 R:= {1}:
%p A339112 for p in PSP do
%p A339112 Rp:= {}:
%p A339112 for k from 1 while p^k <= N do
%p A339112 Rpk:= select(`<=`,R, N/p^k);
%p A339112 Rp:= Rp union map(`*`,Rpk, p^k);
%p A339112 od;
%p A339112 R:= R union Rp;
%p A339112 od:
%p A339112 sort(convert(R,list)); # _Robert Israel_, Nov 03 2024
%t A339112 semiQ[n_]:=PrimeOmega[n]==2;
%t A339112 Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]
%Y A339112 These primes (of semiprime index) are listed by A106349.
%Y A339112 The strict (squarefree) case is A340020.
%Y A339112 The prime instead of semiprime version:
%Y A339112 primes: A006450
%Y A339112 products: A076610
%Y A339112 strict: A302590
%Y A339112 The nonprime instead of semiprime version:
%Y A339112 primes: A007821
%Y A339112 products: A320628
%Y A339112 odd: A320629
%Y A339112 strict: A340104
%Y A339112 odd strict: A340105
%Y A339112 The squarefree semiprime instead of semiprime version:
%Y A339112 strict: A309356
%Y A339112 primes: A322551
%Y A339112 products: A339113
%Y A339112 A001358 lists semiprimes, with odd and even terms A046315 and A100484.
%Y A339112 A006881 lists squarefree semiprimes.
%Y A339112 A037143 lists primes and semiprimes (and 1).
%Y A339112 A056239 gives the sum of prime indices, which are listed by A112798.
%Y A339112 A084126 and A084127 give the prime factors of semiprimes.
%Y A339112 A101048 counts partitions into semiprimes.
%Y A339112 A302242 is the weight of the multiset of multisets with MM-number n.
%Y A339112 A305079 is the number of connected components for MM-number n.
%Y A339112 A320892 lists even-omega non-products of distinct semiprimes.
%Y A339112 A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
%Y A339112 A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
%Y A339112 A338898, A338912, and A338913 give the prime indices of semiprimes.
%Y A339112 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 A339112 Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.
%K A339112 nonn
%O A339112 1,2
%A A339112 _Gus Wiseman_, Mar 12 2021