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 A339741 #24 Jan 09 2021 15:05:46
%S A339741 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A339741 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A339741 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84
%N A339741 Products of distinct primes or squarefree semiprimes.
%C A339741 First differs from A212167 in lacking 1080, with prime indices {1,1,1,2,2,2,3}.
%C A339741 First differs from A335433 in lacking 72 (see example).
%C A339741 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A339741 The following are equivalent characteristics for any positive integer n:
%C A339741 (1) the prime factors of n can be partitioned into distinct singletons and strict pairs, i.e., into a set of half-loops and edges;
%C A339741 (2) n can be factored into distinct primes or squarefree semiprimes;
%C A339741 (3) the prime signature of n is half-loop-graphical.
%H A339741 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H A339741 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%e A339741 The sequence of terms together with their prime indices begins:
%e A339741 1: {} 20: {1,1,3} 39: {2,6}
%e A339741 2: {1} 21: {2,4} 41: {13}
%e A339741 3: {2} 22: {1,5} 42: {1,2,4}
%e A339741 5: {3} 23: {9} 43: {14}
%e A339741 6: {1,2} 26: {1,6} 44: {1,1,5}
%e A339741 7: {4} 28: {1,1,4} 45: {2,2,3}
%e A339741 10: {1,3} 29: {10} 46: {1,9}
%e A339741 11: {5} 30: {1,2,3} 47: {15}
%e A339741 12: {1,1,2} 31: {11} 50: {1,3,3}
%e A339741 13: {6} 33: {2,5} 51: {2,7}
%e A339741 14: {1,4} 34: {1,7} 52: {1,1,6}
%e A339741 15: {2,3} 35: {3,4} 53: {16}
%e A339741 17: {7} 36: {1,1,2,2} 55: {3,5}
%e A339741 18: {1,2,2} 37: {12} 57: {2,8}
%e A339741 19: {8} 38: {1,8} 58: {1,10}
%e A339741 For example, we have 36 = (2*3*6), so 36 is in the sequence. On the other hand, a complete list of all strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72). Since none of these consists of only primes or squarefree semiprimes, 72 is not in the sequence. A complete list of all factorizations of 1080 into primes or squarefree semiprimes is:
%e A339741 (2*2*2*3*3*3*5)
%e A339741 (2*2*2*3*3*15)
%e A339741 (2*2*3*3*3*10)
%e A339741 (2*2*3*3*5*6)
%e A339741 (2*2*3*6*15)
%e A339741 (2*3*3*6*10)
%e A339741 (2*3*5*6*6)
%e A339741 (2*6*6*15)
%e A339741 (3*6*6*10)
%e A339741 (5*6*6*6)
%e A339741 Since none of these is strict, 1080 is not in the sequence.
%t A339741 sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A339741 Select[Range[100],sqps[#]!={}&]
%Y A339741 See link for additional cross-references.
%Y A339741 Allowing only primes gives A013929.
%Y A339741 Not allowing primes gives A339561.
%Y A339741 Complement of A339740.
%Y A339741 Positions of positive terms in A339742.
%Y A339741 Allowing squares of primes gives the complement of A339840.
%Y A339741 Unlabeled multiset partitions of this type are counted by A339888.
%Y A339741 A001055 counts factorizations.
%Y A339741 A001358 lists semiprimes, with squarefree case A006881.
%Y A339741 A002100 counts partitions into squarefree semiprimes.
%Y A339741 A339841 have exactly one factorization into primes or semiprimes.
%Y A339741 Cf. A001221, A005117, A028260, A030229, A050320, A112798, A309356, A320663, A320893, A320924, A338899.
%K A339741 nonn
%O A339741 1,2
%A A339741 _Gus Wiseman_, Dec 23 2020