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A320892
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Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.
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36
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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[#]=={}]&]
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PROG
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(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));
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CROSSREFS
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Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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