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A103399 Semiprimes in A103379. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
REFERENCES
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.
LINKS
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988), 1-16.
FORMULA
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.
EXAMPLE
A103379(21) = 4 = 2 * 2, which is semiprime, hence 4 is in this sequence.
MAPLE
isA103379 := proc(n)
option remember ;
local i ;
for i from 1 do
if A103379(i) = n then
return true ;
elif A103379(i) > n then
return false ;
fi;
od:
end proc:
A103399 := proc(n)
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
printf("%d, \n", A103399(n)) ;
end do: # R. J. Mathar, Aug 30 2008
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A335250 A103396 A103400 * A103398 A103397 A103394
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Feb 15 2005
EXTENSIONS
Corrected from a(15) on by R. J. Mathar, Aug 30 2008
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)