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 A103389 Primes in A103379 (= 11-delayed Fibonacci b(n) = b(n-11)+b(n-12) or = 1 for n<12). 4
 2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 8179, 65657, 68053, 72421, 80429, 258949, 493109, 16399511, 33609887, 34225183, 1387603957, 5575987679, 15932884421, 35689079297, 693128029907, 957136790429, 1129233918343, 10363690074667, 41632551979939, 10815125582078291 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS These primes 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 (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 and k=10 case is A103378. 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 1.062169167864255148458944276143126923146557407121804298167945495796... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/12))^(1/12)))^(1/12))))^(1/12)))))^(1/12))))). This sequence of prime values in this k=11 case is this sequence. The sequence of semiprime values in this k=11 case is A103399. N.B.: a(n) in the above does not refer to the present sequence but to the delayed Fibonacci sequence itself, here A103379. - M. F. Hasler, Sep 19 2015 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 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 A103379 and A000040, 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(20) = 3, which is prime, hence 3 is in this sequence. MAPLE A103379 := proc(n) option remember ; if n <= 12 then 1; else procname(n-11)+procname(n-12) ; fi; end: 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: A103389 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if isprime(a) then if isA103379(a) then RETURN(a) ; fi; fi; od: fi; end: for n from 1 to 37 do printf("%d, ", A103389(n)) ; od: # R. J. Mathar, Aug 30 2008 MATHEMATICA Clear[a]; k11; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103389=Union[Select[Array[a, 1000], PrimeQ]] N[Solve[x^12 - x - 1 == 0, x], 111][] (* Program, edit and extension by Ray Chandler and Robert G. Wilson v, irrelevant code deleted by M. F. Hasler, Sep 19 2015 *) Select[LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 700], PrimeQ]//Union (* Harvey P. Dale, Apr 22 2016 *) PROG (PARI) {a=vector(m=12, n, 1); L=0; for(n=m, 10^5, isprime(a[i=n%m+1]+=a[(n+1)%m+1])&&L

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Last modified May 18 06:33 EDT 2021. Contains 343994 sequences. (Running on oeis4.)