A339112 Products of primes of semiprime index (A106349).

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

Table of n, a(n) for n=1..55.

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)

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).

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

Sequence in context: A273641 A214794 A043104 * A340020 A106349 A293657

Adjacent sequences: A339109 A339110 A339111 * A339113 A339114 A339115

Keyword

nonn

Author

Gus Wiseman, Mar 12 2021

Status

approved