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A320892
Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.
36
16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025
OFFSET
1,1
COMMENTS
A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018
LINKS
EXAMPLE
A complete list of all factorizations of 1296 into semiprimes is:
1296 = (4*4*9*9)
1296 = (4*6*6*9)
1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.
MATHEMATICA
strsemfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strsemfacs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];
Select[Range[1000], And[EvenQ[PrimeOmega[#]], strsemfacs[#]=={}]&]
PROG
(PARI)
A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega, Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs, d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2, 0, (0==A322353(n))); \\ Antti Karttunen, Dec 06 2018
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 23 2018
STATUS
approved