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A339561 Products of distinct squarefree semiprimes.

%I A339561 #12 Feb 16 2025 08:34:01
%S A339561 1,6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,60,62,65,69,74,
%T A339561 77,82,84,85,86,87,90,91,93,94,95,106,111,115,118,119,122,123,126,129,
%U A339561 132,133,134,140,141,142,143,145,146,150,155,156,158,159,161,166
%N A339561 Products of distinct squarefree semiprimes.
%C A339561 First differs from A320911 in lacking 36.
%C A339561 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A339561 The following are equivalent characteristics for any positive integer n:
%C A339561 (1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
%C A339561 (2) n can be factored into distinct squarefree semiprimes;
%C A339561 (3) the prime signature of n is graphical.
%H A339561 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339561 A339561 \/ A320894 = A028260.
%e A339561 The sequence of terms together with their prime indices begins:
%e A339561       1: {}        55: {3,5}         91: {4,6}
%e A339561       6: {1,2}     57: {2,8}         93: {2,11}
%e A339561      10: {1,3}     58: {1,10}        94: {1,15}
%e A339561      14: {1,4}     60: {1,1,2,3}     95: {3,8}
%e A339561      15: {2,3}     62: {1,11}       106: {1,16}
%e A339561      21: {2,4}     65: {3,6}        111: {2,12}
%e A339561      22: {1,5}     69: {2,9}        115: {3,9}
%e A339561      26: {1,6}     74: {1,12}       118: {1,17}
%e A339561      33: {2,5}     77: {4,5}        119: {4,7}
%e A339561      34: {1,7}     82: {1,13}       122: {1,18}
%e A339561      35: {3,4}     84: {1,1,2,4}    123: {2,13}
%e A339561      38: {1,8}     85: {3,7}        126: {1,2,2,4}
%e A339561      39: {2,6}     86: {1,14}       129: {2,14}
%e A339561      46: {1,9}     87: {2,10}       132: {1,1,2,5}
%e A339561      51: {2,7}     90: {1,2,2,3}    133: {4,8}
%e A339561 For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
%e A339561   (6*10*15*77)
%e A339561   (6*10*21*55)
%e A339561   (6*10*33*35)
%e A339561   (6*14*15*55)
%e A339561   (6*15*22*35)
%e A339561   (10*14*15*33)
%e A339561   (10*15*21*22),
%e A339561 so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.
%t A339561 sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
%t A339561 Select[Range[100],sqs[#]!={}&]
%Y A339561 A309356 is a kind of universal embedding.
%Y A339561 A320894 is the complement in A028260.
%Y A339561 A320911 lists all (not just distinct) products of squarefree semiprimes.
%Y A339561 A339560 counts the partitions with these Heinz numbers.
%Y A339561 A339661 has nonzero terms at these positions.
%Y A339561 A001358 lists semiprimes, with squarefree case A006881.
%Y A339561 A005117 lists squarefree numbers.
%Y A339561 A320656 counts factorizations into squarefree semiprimes.
%Y A339561 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339561 - A058696 counts partitions of 2n (A300061).
%Y A339561 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339561 - A209816 counts multigraphical partitions (A320924).
%Y A339561 - A320921 counts connected graphical partitions (A320923).
%Y A339561 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339561 - A339656 counts loop-graphical partitions (A339658).
%Y A339561 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339561 - A000569 counts graphical partitions (A320922).
%Y A339561 The following count partitions of even length and give their Heinz numbers:
%Y A339561 - A027187 has no additional conditions (A028260).
%Y A339561 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339561 - A338914 can be partitioned into strict pairs (A320911).
%Y A339561 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339561 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339561 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339561 - A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
%Y A339561 Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.
%K A339561 nonn
%O A339561 1,2
%A A339561 _Gus Wiseman_, Dec 13 2020