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%I #25 Sep 08 2022 08:46:16
%S 1,3,35,247,1029,3131,7783,16815,32777,59059,100011,161063,248845,
%T 371307,537839,759391,1048593,1419875,1889587,2476119,3200021,4084123,
%U 5153655,6436367,7962649,9765651,11881403,14348935,17210397,20511179,24300031,28629183,33554465
%N a(n) = n^5 + n + 1.
%C For n>1 these are odd composite numbers: all terms a(n) are divisible by number h(n) = GCD(n^5+n+1,(n+1)^5+n) = GCD(a(n), a(n+1)-2) = (n*(n+1)+1)*GCD(n*(n+1)-1, 5) where 1 < h(n) < a(n) for all n>1. Sequence of corresponding numbers h(n) for n>1: 35, 13, 21, 31, 43, 285, ... For example, a(7) = 16815 is divisible by number h(7) = (7*(7+1)+1)*GCD(7*(7+1)-1, 5) = 57*GCD(55, 5) = 57*5 = 285.
%C We name a set of k sequences IOPR_k(n) = {a_1(n) = a(n), a_2(n) = a(n) + 2, ..., a_k(n) = a(n) + 2*(k - 1)} as infinite nonprime k-lane road if a arithmetic function a(n) defined by arithmetic operations produces for all n > h (h = a small integer >= 0) odd terms such that all values a(n), a(n) + 2, ..., a(n) + 2*(k - 1) are composites. We say sequences a_1(n) = a(n), a_2(n) = a(n) + 2, ..., a_k(n) = a(n) + 2*(k - 1) are k-th lanes of set IOPR_k(n).
%C For example, sequence A016945(n) = 6*n + 3 = IOPR_1(n) for k=1.
%C This sequence a(n) is 2nd lane of set of sequences IOPR_2(n) = {a_1(n) = A271208(n) = a(n) - 2 = n^5 + n - 1, a_2(n) = a(n) = n^5 + n + 1}.
%C If p = prime > 2 of the form 3m - 1 from A003627 then sets of 2 sequences {n^p + n - 1, n^p + n + 1} = IOPR_2(n) for all p.
%C Also sets of 2 sequences {n^k + n - 1, n^k + n + 1} = IOPR_2(n) for all k>2 from A016789.
%C In general, if k>2 is number of the form 3m - 1 from A016789 then sequences a(n) = n^k + n - 1 and b(n) = a(n) + 2 = n^k + n + 1 produces for all n > 1 odd composite terms. The terms of sequence a(n) = n^k + n - 1 are divisible for all n > 1 by number h(n) = GCD(n^k+n-1,(n-1)^k+n) = GCD(a(n), a(n-1)+2) = (n*(n-1)+1)*GCD(n*(n-1)-1, k) where 1 < h(n) < a(n) for all n>1. The terms of sequence b(n) = a(n) + 2 = n^k + n + 1 are divisible for all n > 1 by number h(n) = GCD(n^k+n+1,(n+1)^k+n) = GCD(a(n), a(n+1)-2) = (n*(n+1)+1)*GCD(n*(n+1)-1, k) where 1 < h(n) < a(n) for all n>1.
%C Are there any sets of sequences IOPR_k(n) for k>2? For example, like set of sequences {A161945(n), A161945(n) + 2, A161945(n) + 4} is not an infinite nonprime 3-lane road because sequence A161945 is not defined by arithmetic operations.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = A271208(n) + 2.
%F From _Wesley Ivan Hurt_, Apr 02 2016: (Start)
%F G.f.: (1-3*x+32*x^2+62*x^3+27*x^4+x^5) / (x-1)^6.
%F a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6), n>5. (End)
%F a(n) = A131471(n) + 1. - _Omar E. Pol_, Apr 05 2016
%p A271209:=n->n^5 + n + 1: seq(A271209(n), n=0..40); # _Wesley Ivan Hurt_, Apr 02 2016
%t Table[n^5+n+1, {n, 0, 100}] (* _Waldemar Puszkarz_, Apr 02, 2016 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,3,35,247,1029,3131},40] (* _Harvey P. Dale_, Jul 24 2016 *)
%o (Magma) [n^5+n+1: n in[0..100]]
%o (PARI) for(n=0, 100, print1(n^5+n+1, ", ")) \\ _Waldemar Puszkarz_, Apr 02 2016
%Y Cf. A003627, A016789, A131471, A161945, A271208.
%K nonn,easy
%O 0,2
%A _Jaroslav Krizek_, Apr 02 2016
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