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Products of distinct squarefree semiprimes.
25

%I #11 Dec 18 2020 07:59:11

%S 1,6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,58,60,62,65,69,74,

%T 77,82,84,85,86,87,90,91,93,94,95,106,111,115,118,119,122,123,126,129,

%U 132,133,134,140,141,142,143,145,146,150,155,156,158,159,161,166

%N Products of distinct squarefree semiprimes.

%C First differs from A320911 in lacking 36.

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

%C The following are equivalent characteristics for any positive integer n:

%C (1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);

%C (2) n can be factored into distinct squarefree semiprimes;

%C (3) the prime signature of n is graphical.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>

%F A339561 \/ A320894 = A028260.

%e The sequence of terms together with their prime indices begins:

%e 1: {} 55: {3,5} 91: {4,6}

%e 6: {1,2} 57: {2,8} 93: {2,11}

%e 10: {1,3} 58: {1,10} 94: {1,15}

%e 14: {1,4} 60: {1,1,2,3} 95: {3,8}

%e 15: {2,3} 62: {1,11} 106: {1,16}

%e 21: {2,4} 65: {3,6} 111: {2,12}

%e 22: {1,5} 69: {2,9} 115: {3,9}

%e 26: {1,6} 74: {1,12} 118: {1,17}

%e 33: {2,5} 77: {4,5} 119: {4,7}

%e 34: {1,7} 82: {1,13} 122: {1,18}

%e 35: {3,4} 84: {1,1,2,4} 123: {2,13}

%e 38: {1,8} 85: {3,7} 126: {1,2,2,4}

%e 39: {2,6} 86: {1,14} 129: {2,14}

%e 46: {1,9} 87: {2,10} 132: {1,1,2,5}

%e 51: {2,7} 90: {1,2,2,3} 133: {4,8}

%e 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 (6*10*15*77)

%e (6*10*21*55)

%e (6*10*33*35)

%e (6*14*15*55)

%e (6*15*22*35)

%e (10*14*15*33)

%e (10*15*21*22),

%e 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 sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];

%t Select[Range[100],sqs[#]!={}&]

%Y A309356 is a kind of universal embedding.

%Y A320894 is the complement in A028260.

%Y A320911 lists all (not just distinct) products of squarefree semiprimes.

%Y A339560 counts the partitions with these Heinz numbers.

%Y A339661 has nonzero terms at these positions.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A005117 lists squarefree numbers.

%Y A320656 counts factorizations into squarefree semiprimes.

%Y The following count vertex-degree partitions and give their Heinz numbers:

%Y - A058696 counts partitions of 2n (A300061).

%Y - A000070 counts non-multigraphical partitions of 2n (A339620).

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A320921 counts connected graphical partitions (A320923).

%Y - A339655 counts non-loop-graphical partitions of 2n (A339657).

%Y - A339656 counts loop-graphical partitions (A339658).

%Y - A339617 counts non-graphical partitions of 2n (A339618).

%Y - A000569 counts graphical partitions (A320922).

%Y The following count partitions of even length and give their Heinz numbers:

%Y - A027187 has no additional conditions (A028260).

%Y - A096373 cannot be partitioned into strict pairs (A320891).

%Y - A338914 can be partitioned into strict pairs (A320911).

%Y - A338915 cannot be partitioned into distinct pairs (A320892).

%Y - A338916 can be partitioned into distinct pairs (A320912).

%Y - A339559 cannot be partitioned into distinct strict pairs (A320894).

%Y - A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).

%Y Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 13 2020