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A271209
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a(n) = n^5 + n + 1.
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3
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1, 3, 35, 247, 1029, 3131, 7783, 16815, 32777, 59059, 100011, 161063, 248845, 371307, 537839, 759391, 1048593, 1419875, 1889587, 2476119, 3200021, 4084123, 5153655, 6436367, 7962649, 9765651, 11881403, 14348935, 17210397, 20511179, 24300031, 28629183, 33554465
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OFFSET
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0,2
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COMMENTS
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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.
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).
For example, sequence A016945(n) = 6*n + 3 = IOPR_1(n) for k=1.
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}.
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.
Also sets of 2 sequences {n^k + n - 1, n^k + n + 1} = IOPR_2(n) for all k>2 from A016789.
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.
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.
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LINKS
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FORMULA
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G.f.: (1-3*x+32*x^2+62*x^3+27*x^4+x^5) / (x-1)^6.
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)
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MAPLE
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MATHEMATICA
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LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 3, 35, 247, 1029, 3131}, 40] (* Harvey P. Dale, Jul 24 2016 *)
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PROG
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(Magma) [n^5+n+1: n in[0..100]]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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