A339113 Products of primes of squarefree semiprime index (A322551).

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937

Offset: 1

Author

Gus Wiseman, Mar 12 2021

Keywords

• nonn

Comments

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also MM-numbers of labeled multigraphs (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}}.

Examples

			The sequence of terms together with the corresponding multigraphs begins:
      1: {}               233: {{2,7}}          487: {{2,11}}
     13: {{1,2}}          257: {{3,5}}          491: {{1,15}}
     29: {{1,3}}          269: {{2,8}}          499: {{3,8}}
     43: {{1,4}}          271: {{1,10}}         559: {{1,2},{1,4}}
     47: {{2,3}}          293: {{1,11}}         577: {{1,16}}
     73: {{2,4}}          313: {{3,6}}          607: {{2,12}}
     79: {{1,5}}          347: {{2,9}}          611: {{1,2},{2,3}}
    101: {{1,6}}          373: {{1,12}}         631: {{3,9}}
    137: {{2,5}}          377: {{1,2},{1,3}}    647: {{1,17}}
    139: {{1,7}}          389: {{4,5}}          653: {{4,7}}
    149: {{3,4}}          421: {{1,13}}         673: {{1,18}}
    163: {{1,8}}          439: {{3,7}}          677: {{2,13}}
    167: {{2,6}}          443: {{1,14}}         727: {{2,14}}
    169: {{1,2},{1,2}}    449: {{2,10}}         751: {{4,8}}
    199: {{1,9}}          467: {{4,6}}          757: {{1,19}}
		

Crossrefs

These primes (of squarefree semiprime index) are listed by A322551.

The strict (squarefree) case is A309356.

The prime instead of squarefree semiprime version:

primes: A006450

products: A076610

strict: A302590

The nonprime instead of squarefree semiprime version:

primes: A007821

products: A320628

odd: A320629

strict: A340104

odd strict: A340105

The semiprime instead of squarefree semiprime version:

primes: A106349

products: A339112

strict: A340020

A001358 lists semiprimes, with odd/even terms A046315/A100484.

A002100 counts partitions into squarefree semiprimes.

A005117 lists squarefree numbers.

A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.

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

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

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

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

A338899/A270650/A270652 give the prime indices of squarefree semiprimes.

A339561 lists products of distinct squarefree semiprimes (ranking: A339560).

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, A289509, A320461.

Programs

Mathematica

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]