OFFSET
1,1
COMMENTS
LINKS
Eric Weisstein's World of Mathematics, Degree Sequence.
EXAMPLE
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.
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[#]!={}&]
CROSSREFS
See link for additional cross references.
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.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 30 2020
STATUS
approved