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A339112
Products of primes of semiprime index (A106349).
14
1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647
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
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
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
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.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
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).
Sequence in context: A273641 A214794 A043104 * A340020 A106349 A293657
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Mar 12 2021
STATUS
approved