

A103389


Primes in A103379 (= 11delayed Fibonacci b(n) = b(n11)+b(n12) 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
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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(nk) + a(n[k+1]). For this k=11 case, the ratio of successive terms a(n)/a(n1) 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) 229245.


LINKS

Table of n, a(n) for n=1..35.
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) 287302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988), 116.


FORMULA

Intersection of A103379 and A000040, where A103379 is: for n>12: a(n) = a(n11) + a(n12). 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(n11)+procname(n12) ; 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(n1)+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[nk]+a[nk1]; A103389=Union[Select[Array[a, 1000], PrimeQ]] N[Solve[x^12  x  1 == 0, x], 111][[2]] (* 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<a[i]&&print1(L=a[i], ", "))} \\ M. F. Hasler, Sep 19 2015


CROSSREFS

Cf. A000040, A000931, A079398, A103372A103381, A103379, A103399.
Sequence in context: A070805 A255161 A103385 * A103387 A103388 A103386
Adjacent sequences: A103386 A103387 A103388 * A103390 A103391 A103392


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Feb 15 2005


EXTENSIONS

Corrected from a(16) on by R. J. Mathar, Aug 30 2008
Edited and more terms added by M. F. Hasler, Sep 19 2015


STATUS

approved



