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 A103400 Semiprimes in A103380. 2
 4, 9, 15, 21, 33, 38, 58, 65, 86, 106, 121, 129, 265, 511, 2047, 2049, 4097, 4109, 8193, 17855, 19857, 34709, 66233, 104739, 130953, 131209, 140474, 220918, 258931, 511673, 540951 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These semiprimes are elements of the k=12 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, k=5 case is A103373, k=6 case is A103374, k=7 case is A103375, k=8 case is A103376, k=9 case is A103377, k=10 case is A103378 and k=11 case is A103379. 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=12 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^13 - x - 1 = 0. This is the real constant 1.0570505752212283848816867278047539300461075... . 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=12 case is A103390; This sequence of semiprime values in this k=12 case is A103400. REFERENCES A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245. LINKS Table of n, a(n) for n=1..31. J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms Richard Padovan, Dom Hans van der Laan and the Plastic Number. 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 A103380 and A001358, where A103380 is: for n>13: a(n) = a(n-12) + a(n-13). 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. MAPLE A103380 := proc(n) option remember ; if n <= 13 then 1; else procname(n-12)+procname(n-13) ; fi; end: isA103380 := proc(n) option remember ; local i ; for i from 1 do if A103380(i) = n then RETURN(true) ; elif A103380(i) > n then RETURN(false) ; fi; od: end: A103400 := 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 isA103380(a) then RETURN(a) ; fi; fi; od: fi; end: for n from 1 to 37 do printf("%d, ", A103400(n)) ; od: # R. J. Mathar, Aug 30 2008 MATHEMATICA SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k12; 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 Cf. A001358, A000931, A079398, A103372-103381, A103380, A103390. Sequence in context: A162801 A335250 A103396 * A103399 A103398 A103397 Adjacent sequences: A103397 A103398 A103399 * A103401 A103402 A103403 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Feb 16 2005 EXTENSIONS Corrected from a(15) on by R. J. Mathar, Aug 30 2008 STATUS approved

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Last modified March 5 02:31 EST 2024. Contains 370537 sequences. (Running on oeis4.)