login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A103388 Primes in A103378. 3

%I #17 Sep 19 2015 23:49:55

%S 2,3,5,7,17,31,71,127,157,227,257,293,349,419,503,32299,33343,72421,

%T 80429,134269,140473,252761,2499061,201329923,607488611,1005428989,

%U 2920552289,8185638173,10676478541,14058719281,15985335181,34020175663,159315910211,1448256661853

%N Primes in A103378.

%C 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.

%C N.B.: a(n) in the above does not refer to the terms of this sequence. - _M. F. Hasler_, Sep 19 2015

%F 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).

%p 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

%t 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, . *)

%o (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

%Y Cf. A000045, A000931, A079398, A103372-A103381, A103378, A103398.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 15 2005

%E Corrected from a(16) on by _R. J. Mathar_, Aug 30 2008

%E Edited and more terms added by _M. F. Hasler_, Sep 19 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)