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A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.

%I A340017 #13 Dec 30 2020 19:57:48
%S A340017 36,100,196,216,225,360,441,484,504,540,600,676,756,792,936,1000,1089,
%T A340017 1156,1176,1188,1224,1225,1296,1350,1368,1400,1404,1444,1500,1521,
%U A340017 1656,1836,1960,2052,2088,2116,2160,2200,2232,2250,2484,2600,2601,2646,2664,2744
%N A340017 Products of squarefree semiprimes that are not products of distinct squarefree semiprimes.
%C A340017 Of course, every number is a product of squarefree numbers (A050320).
%C A340017 A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
%C A340017 All terms have even Omega (A001222, A028260).
%H A340017 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H A340017 Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F A340017 Equals A320894 /\ A320911.
%F A340017 Numbers n such that A320656(n) > 0 but A339661(n) = 0.
%e A340017 The sequence of terms together with their prime indices begins:
%e A340017       36: {1,1,2,2}        1000: {1,1,1,3,3,3}
%e A340017      100: {1,1,3,3}        1089: {2,2,5,5}
%e A340017      196: {1,1,4,4}        1156: {1,1,7,7}
%e A340017      216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
%e A340017      225: {2,2,3,3}        1188: {1,1,2,2,2,5}
%e A340017      360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
%e A340017      441: {2,2,4,4}        1225: {3,3,4,4}
%e A340017      484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
%e A340017      504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
%e A340017      540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
%e A340017      600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
%e A340017      676: {1,1,6,6}        1404: {1,1,2,2,2,6}
%e A340017      756: {1,1,2,2,2,4}    1444: {1,1,8,8}
%e A340017      792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
%e A340017      936: {1,1,1,2,2,6}    1521: {2,2,6,6}
%e A340017 For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
%e A340017   7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
%e A340017 but since none of these is strict, 7560 is in the sequence.
%t A340017 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 A340017 Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]
%Y A340017 See link for additional cross references.
%Y A340017 The distinct prime shadows (under A181819) of these terms are A339842.
%Y A340017 Factorizations into squarefree semiprimes are counted by A320656.
%Y A340017 Products of squarefree semiprimes that are not products of distinct semiprimes are A320893.
%Y A340017 Factorizations into distinct squarefree semiprimes are A339661.
%Y A340017 For the next four lines, we list numbers with even Omega (A028260).
%Y A340017 - A320891 cannot be factored into squarefree semiprimes.
%Y A340017 - A320894 cannot be factored into distinct squarefree semiprimes.
%Y A340017 - A320911 can be factored into squarefree semiprimes.
%Y A340017 - A339561 can be factored into distinct squarefree semiprimes.
%Y A340017 A001358 lists semiprimes, with squarefree case A006881.
%Y A340017 A002100 counts partitions into squarefree semiprimes.
%Y A340017 A030229 lists squarefree numbers with even Omega.
%Y A340017 A050320 counts factorizations into squarefree numbers.
%Y A340017 A050326 counts factorizations into distinct squarefree numbers.
%Y A340017 A181819 is the Heinz number of the prime signature of n (prime shadow).
%Y A340017 A320656 counts factorizations into squarefree semiprimes.
%Y A340017 A339560 can be partitioned into distinct strict pairs.
%Y A340017 Cf. A001055, A001222, A005117, A007717, A096373, A112798, A300061, A322353, A339559, A338899, A339740.
%K A340017 nonn
%O A340017 1,1
%A A340017 _Gus Wiseman_, Dec 30 2020