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4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 129, 265, 511, 2047, 2049, 4097, 4109, 17855, 19857, 32663, 34709, 104739, 130393, 131889, 140474, 220918, 262978, 266174, 274759, 540933, 568083, 1312526, 1665242, 1833203, 2179101, 2295571
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OFFSET
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1,1
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COMMENTS
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These semiprimes are elements of the k=11 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=11 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^12 - x - 1 = 0. This is the real constant 1.06216916786425514845894427614312692314655740712180429816794549579... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/12))^(1/12)))^(1/12))))^(1/12)))))^(1/12))))). The sequence of prime values in this k=11 case is A103389; This sequence of semiprime values in this k=11 case is this sequence.
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REFERENCES
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A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (1999) 229-245.
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LINKS
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FORMULA
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Intersection of A103379 and A001358, where A103379 is: for n>12: a(n) = a(n-11) + a(n-12). a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) = 1.
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EXAMPLE
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A103379(21) = 4 = 2 * 2, which is semiprime, hence 4 is in this sequence.
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MAPLE
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isA103379 := proc(n)
option remember ;
local i ;
for i from 1 do
return true ;
return false ;
fi;
od:
end proc:
option remember ;
local a, i ;
if n = 1 then
4;
else
for a from procname(n-1)+1 do
if numtheory[bigomega](a) = 2 then
if isA103379(a) then
return a ;
fi;
fi;
end do:
end if;
end proc:
for n from 1 do
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MATHEMATICA
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SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k11; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103379=Array[a, 100] A103389=Union[Select[Array[a, 1000], PrimeQ]] A103399=Union[Select[Array[a, 300], SemiprimeQ]] N[Solve[x^12 - x - 1 == 0, x], 111][[2]] (* Program, edit and extension by Ray Chandler and Robert G. Wilson v *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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