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A339840
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Numbers that cannot be factored into distinct primes or semiprimes.
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7
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16, 32, 64, 81, 96, 128, 160, 192, 224, 243, 256, 288, 320, 352, 384, 416, 448, 486, 512, 544, 576, 608, 625, 640, 704, 729, 736, 768, 800, 832, 864, 896, 928, 960, 972, 992, 1024, 1088, 1152, 1184, 1215, 1216, 1280, 1312, 1344, 1376, 1408, 1458, 1472, 1504
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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 prime numbers.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
16: {1,1,1,1}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
81: {2,2,2,2}
96: {1,1,1,1,1,2}
128: {1,1,1,1,1,1,1}
160: {1,1,1,1,1,3}
192: {1,1,1,1,1,1,2}
224: {1,1,1,1,1,4}
243: {2,2,2,2,2}
256: {1,1,1,1,1,1,1,1}
288: {1,1,1,1,1,2,2}
320: {1,1,1,1,1,1,3}
352: {1,1,1,1,1,5}
384: {1,1,1,1,1,1,1,2}
416: {1,1,1,1,1,6}
448: {1,1,1,1,1,1,4}
486: {1,2,2,2,2,2}
For example, a complete list of all factorizations of 192 into primes or semiprimes is:
(2*2*2*2*2*2*3)
(2*2*2*2*2*6)
(2*2*2*2*3*4)
(2*2*2*4*6)
(2*2*3*4*4)
(2*4*4*6)
(3*4*4*4)
Since none of these is strict, 192 is in the sequence.
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MAPLE
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filter:= proc(n)
g(map(t -> t[2], ifactors(n)[2]))
end proc;
g:= proc(L) option remember; local x, i, j, t, s, Cons, R;
if nops(L) = 1 then return L[1] > 3
elif nops(L) = 2 then return max(L) > 4
fi;
Cons:= {seq(x[i] + x[i, i] + add(x[j, i], j=1..i-1)
+ add(x[i, j], j=i+1..nops(L)) = L[i], i=1..nops(L))};
R:= traperror(Optimization:-LPSolve(0, Cons, assume=binary));
type(R, string)
end proc:
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[1000], Select[facs[#], UnsameQ@@#&&SubsetQ[{1, 2}, PrimeOmega/@#]&]=={}&]
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CROSSREFS
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Allowing only primes gives A013929.
Removing all squares of primes gives A339740.
These are the positions of zeros in A339839.
A002100 counts partitions into squarefree semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
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.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339742 into distinct primes or 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.
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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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