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

36, 100, 196, 216, 225, 360, 441, 484, 504, 540, 600, 676, 756, 792, 936, 1000, 1089, 1156, 1176, 1188, 1224, 1225, 1296, 1350, 1368, 1400, 1404, 1444, 1500, 1521, 1656, 1836, 1960, 2052, 2088, 2116, 2160, 2200, 2232, 2250, 2484, 2600, 2601, 2646, 2664, 2744

Offset: 1

Author

Gus Wiseman, Dec 30 2020

Keywords

• nonn

Comments

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

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

All terms have even Omega (A001222, A028260).

Examples

			The sequence of terms together with their prime indices begins:
      36: {1,1,2,2}        1000: {1,1,1,3,3,3}
     100: {1,1,3,3}        1089: {2,2,5,5}
     196: {1,1,4,4}        1156: {1,1,7,7}
     216: {1,1,1,2,2,2}    1176: {1,1,1,2,4,4}
     225: {2,2,3,3}        1188: {1,1,2,2,2,5}
     360: {1,1,1,2,2,3}    1224: {1,1,1,2,2,7}
     441: {2,2,4,4}        1225: {3,3,4,4}
     484: {1,1,5,5}        1296: {1,1,1,1,2,2,2,2}
     504: {1,1,1,2,2,4}    1350: {1,2,2,2,3,3}
     540: {1,1,2,2,2,3}    1368: {1,1,1,2,2,8}
     600: {1,1,1,2,3,3}    1400: {1,1,1,3,3,4}
     676: {1,1,6,6}        1404: {1,1,2,2,2,6}
     756: {1,1,2,2,2,4}    1444: {1,1,8,8}
     792: {1,1,1,2,2,5}    1500: {1,1,2,3,3,3}
     936: {1,1,1,2,2,6}    1521: {2,2,6,6}
For example, a complete list of all factorizations of 7560 into squarefree semiprimes is:
  7560 = (6*6*6*35) = (6*6*10*21) = (6*6*14*15),
but since none of these is strict, 7560 is in the sequence.
		

Links

• Eric Weisstein's World of Mathematics, Degree Sequence.
• Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Crossrefs

See link for additional cross references.

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

Factorizations into squarefree semiprimes are counted by A320656.

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

Factorizations into distinct squarefree semiprimes are A339661.

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

- A320891 cannot be factored into squarefree semiprimes.

- A320894 cannot be factored into distinct squarefree semiprimes.

- A320911 can be factored into squarefree semiprimes.

- A339561 can be factored into distinct squarefree semiprimes.

A001358 lists semiprimes, with squarefree case A006881.

A002100 counts partitions into squarefree semiprimes.

A030229 lists squarefree numbers with even Omega.

A050320 counts factorizations into squarefree numbers.

A050326 counts factorizations into distinct squarefree numbers.

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

A320656 counts factorizations into squarefree semiprimes.

A339560 can be partitioned into distinct strict pairs.

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

Programs

Mathematica

strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Select[Range[1000],Select[strr[#],UnsameQ@@#&]=={}&&strr[#]!={}&]

Formula

Equals A320894 /\ A320911.

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