A339740 Non-products of distinct primes or squarefree semiprimes.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Author

Gus Wiseman, Dec 20 2020

Keywords

• nonn

Comments

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

Examples

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.
		

Crossrefs

A013929 allows only primes.

A320894 does not allow primes (but omega is assumed even).

A339741 is the complement.

A339742 has zeros at these positions.

A339840 allows squares of primes.

A001358 lists semiprimes, with squarefree case A006881.

A002100 counts partitions into squarefree semiprimes.

A320663 counts non-isomorphic multiset partitions into singletons or pairs.

A339841 have exactly one factorization into primes or semiprimes.

The following count factorizations:

- A001055 into all positive integers > 1.

- A050326 into distinct squarefree numbers.

- A320655 into semiprimes.

- A320656 into squarefree semiprimes.

- A320732 into primes or semiprimes.

- A322353 into distinct semiprimes.

- A339661 into distinct squarefree semiprimes.

- A339839 into distinct primes or semiprimes.

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

- A058696 counts partitions of 2n (A300061).

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

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

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

- A321728 is conjectured to count non-half-loop-graphical partitions of n.

The following count partitions/factorizations of even length and give their Heinz numbers:

- A027187/A339846 counts all of even length (A028260).

- A096373/A339737 cannot be partitioned into strict pairs (A320891).

- A338915/A339662 cannot be partitioned into distinct pairs (A320892).

- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).

Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

Programs

Mathematica

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqps[#]=={}&]