OFFSET
1,1
COMMENTS
EXAMPLE
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.
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[#]=={}&]
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.
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:
- 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:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 20 2020
STATUS
approved