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2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 32299, 33343, 72421, 80429, 134269, 140473, 252761, 2499061, 201329923, 607488611, 1005428989, 2920552289, 8185638173, 10676478541, 14058719281, 15985335181, 34020175663, 159315910211, 1448256661853
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OFFSET
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1,1
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COMMENTS
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These are the unique primes that are found in the k=10 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=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=10 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^11 - x - 1 = 0. This is the real constant 1.068297188920841276369429588323878282093631016920833444507611946647... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/11))^(1/11)))^(1/11))))^(1/11)))))^(1/11))))). This sequence of prime values in this k=10 case is A103388. The sequence of semiprime values in this k=10 case is A103398.
N.B.: a(n) in the above does not refer to the terms of this sequence. - M. F. Hasler, Sep 19 2015
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LINKS
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FORMULA
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Intersection of A103378 with A000040. A103378 is defined: a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = 1 and for n>11: a(n) = a(n-10) + a(n-11).
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MAPLE
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A103378 := proc(n) option remember ; if n <= 11 then 1; else procname(n-10)+procname(n-11) ; fi; end: isA103378 := proc(n) option remember ; local i ; for i from 1 do if A103378(i) = n then RETURN(true) ; elif A103378(i) > n then RETURN(false) ; fi; od: end: A103388 := proc(n) option remember ; local a; if n = 1 then 2; else a := nextprime(procname(n-1)) ; while true do if isA103378(a) then RETURN(a) ; fi; a := nextprime(a) ; od: fi; end: for n from 1 to 37 do printf("%d, ", A103388(n)) ; od: # R. J. Mathar, Aug 30 2008
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MATHEMATICA
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Clear[a]; k=10; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103387=Union[Select[Array[a, 1000], PrimeQ]] (* See A103377 and A103397 for code related to those. - M. F. Hasler, Sep 19 2015, . *)
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PROG
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(PARI) {a=vector(m=10, 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
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CROSSREFS
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KEYWORD
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easy,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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