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A339113
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Products of primes of squarefree semiprime index (A322551).
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19
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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
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OFFSET
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1,2
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COMMENTS
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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}}.
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LINKS
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EXAMPLE
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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}}
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MATHEMATICA
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sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000], FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; !sqfsemiQ[PrimePi[p]]]&]
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CROSSREFS
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These primes (of squarefree semiprime index) are listed by A322551.
The strict (squarefree) case is A309356.
The prime instead of squarefree semiprime version:
The nonprime instead of squarefree semiprime version:
The semiprime instead of squarefree semiprime version:
A002100 counts partitions into squarefree 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.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
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).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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