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A179939
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Largest semiprime divisor of all composite numbers between semiprime(n) and semiprime(n+1), or 0 if there are none.
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1
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0, 4, 0, 6, 0, 10, 0, 6, 0, 15, 0, 0, 9, 0, 22, 6, 25, 26, 14, 0, 15, 21, 34, 35, 38, 39, 21, 0, 0, 22, 46, 0, 0, 51, 55, 57, 58, 0, 15, 0, 0, 62, 65, 0, 69, 0, 0, 9, 0, 77, 39, 0, 10, 82, 21, 87, 0, 91, 46, 93, 95, 65, 0, 0, 51, 0, 69, 106
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OFFSET
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1,2
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COMMENTS
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This is to A052248 as semiprimes (A001358) are to primes (A000040). This defines a mapping f from semiprimes to semiprimes or 0 and f(s) < s holds for all semiprimes s. There is a block of k-1 consecutive 0's corresponding to each block of k consecutive semiprimes (i.e., a block of two consecutive 0's starting at the least of the triples in A115394).
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 0 because there are no composite numbers between the 1st semiprime 4 and the 2nd semiprime 6.
a(2) = 4 because the composite numbers between the 2nd semiprime 6 and the 3rd semiprime 9 are {8} which is divisible by the semiprime 4=2*2.
a(10) = 15 because the composite numbers between the 10th semiprime 26 and the 11th semiprime 33 are {27, 28, 30, 32} of which the maximum is found for 30 which is divisible by the semiprime 15=3*5.
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MAPLE
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b:= proc(n) option remember; local k;
if n=1 then 4
else for k from b(n-1)+1 while
isprime(k) or add(i[2], i=ifactors(k)[2])<>2
do od; k
fi
end:
a:= proc(n) option remember; local k, l;
k, l:= b(n)+1, b(n+1)-1;
max(0, seq(seq(`if`(irem(j, b(i))=0, b(i), NULL),
i=1..n), j=k..l))
end:
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MATHEMATICA
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(* First run the program for A105999 *) semiPrimeQ[x_] := TrueQ[Plus @@ Last /@ FactorInteger[x] == 2]; spGPF[start_, end_] := Module[{divList, spList}, divList = Union[Flatten[Table[Divisors[n], {n, start + 1, end - 1}]]]; spList = Select[divList, semiPrimeQ]; If[Length[spList] > 0, Return[Max[spList]], Return[0]]]; Table[spGPF[SemiPrime[n], SemiPrime[n + 1]], {n, 50}] (* Alonso del Arte, Jan 13 2011 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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