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Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.
1

%I #13 Dec 30 2020 19:57:48

%S 36,100,196,216,225,360,441,484,504,540,600,676,756,792,936,1000,1089,

%T 1156,1176,1188,1224,1225,1296,1350,1368,1400,1404,1444,1500,1521,

%U 1656,1836,1960,2052,2088,2116,2160,2200,2232,2250,2484,2600,2601,2646,2664,2744

%N Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

%C Of course, every number is a product of squarefree numbers (A050320).

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

%C All terms have even Omega (A001222, A028260).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>

%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>

%F Equals A320894 /\ A320911.

%F Numbers n such that A320656(n) > 0 but A339661(n) = 0.

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

%e 36: {1,1,2,2} 1000: {1,1,1,3,3,3}

%e 100: {1,1,3,3} 1089: {2,2,5,5}

%e 196: {1,1,4,4} 1156: {1,1,7,7}

%e 216: {1,1,1,2,2,2} 1176: {1,1,1,2,4,4}

%e 225: {2,2,3,3} 1188: {1,1,2,2,2,5}

%e 360: {1,1,1,2,2,3} 1224: {1,1,1,2,2,7}

%e 441: {2,2,4,4} 1225: {3,3,4,4}

%e 484: {1,1,5,5} 1296: {1,1,1,1,2,2,2,2}

%e 504: {1,1,1,2,2,4} 1350: {1,2,2,2,3,3}

%e 540: {1,1,2,2,2,3} 1368: {1,1,1,2,2,8}

%e 600: {1,1,1,2,3,3} 1400: {1,1,1,3,3,4}

%e 676: {1,1,6,6} 1404: {1,1,2,2,2,6}

%e 756: {1,1,2,2,2,4} 1444: {1,1,8,8}

%e 792: {1,1,1,2,2,5} 1500: {1,1,2,3,3,3}

%e 936: {1,1,1,2,2,6} 1521: {2,2,6,6}

%e For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:

%e 7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),

%e but since none of these is strict, 7560 is in the sequence.

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

%t Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

%Y See link for additional cross references.

%Y The distinct prime shadows (under A181819) of these terms are A339842.

%Y Factorizations into squarefree semiprimes are counted by A320656.

%Y Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.

%Y Factorizations into distinct squarefree semiprimes are A339661.

%Y For the next four lines, we list numbers with even Omega (A028260).

%Y - A320891 cannot be factored into squarefree semiprimes.

%Y - A320894 cannot be factored into distinct squarefree semiprimes.

%Y - A320911 can be factored into squarefree semiprimes.

%Y - A339561 can be factored into distinct squarefree semiprimes.

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A002100 counts partitions into squarefree semiprimes.

%Y A030229 lists squarefree numbers with even Omega.

%Y A050320 counts factorizations into squarefree numbers.

%Y A050326 counts factorizations into distinct squarefree numbers.

%Y A181819 is the Heinz number of the prime signature of n (prime shadow).

%Y A320656 counts factorizations into squarefree semiprimes.

%Y A339560 can be partitioned into distinct strict pairs.

%Y Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.

%K nonn

%O 1,1

%A _Gus Wiseman_, Dec 30 2020