A161945 -id:A161945 - cp's OEIS frontend

A271209 a(n) = n^5 + n + 1.

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

Offset: 0

Jaroslav Krizek, Apr 02 2016

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.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A003627, A016789, A131471, A161945, A271208.

Magma
```
[n^5+n+1: n in[0..100]];
```

A271209:=n->n^5 + n + 1: seq(A271209(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016

Table[n^5+n+1, {n, 0, 100}] (* Waldemar Puszkarz, Apr 02 2016 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,3,35,247,1029,3131},40] (* Harvey P. Dale, Jul 24 2016 *)

for(n=0, 100, print1(n^5+n+1, ", ")) \\ Waldemar Puszkarz, Apr 02 2016

a(n) = A271208(n) + 2.
From Wesley Ivan Hurt, Apr 02 2016: (Start)
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)
a(n) = A131471(n) + 1. - Omar E. Pol, Apr 05 2016

A271208 a(n) = n^5 + n - 1.

Original entry on oeis.org

-1, 1, 33, 245, 1027, 3129, 7781, 16813, 32775, 59057, 100009, 161061, 248843, 371305, 537837, 759389, 1048591, 1419873, 1889585, 2476117, 3200019, 4084121, 5153653, 6436365, 7962647, 9765649, 11881401, 14348933, 17210395, 20511177, 24300029, 28629181, 33554463

Offset: 0

Jaroslav Krizek, Apr 02 2016

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A003627, A016789, A131471, A161945, A271209.

Magma
```
[n^5+n-1: n in [0..100]];
```

A271208:=n->n^5 + n - 1: seq(A271208(n), n=0..40); # Wesley Ivan Hurt, Apr 02 2016

Table[n^5+n-1, {n, 0, 100}] (* Waldemar Puszkarz, Apr 02 2016 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{-1,1,33,245,1027,3129},40] (* Harvey P. Dale, Sep 02 2020 *)

for(n=0, 100, print1(n^5+n-1, ", ")) \\ Waldemar Puszkarz, Apr 02 2016

a(n) = A271209(n) - 2.
From Wesley Ivan Hurt, Apr 02 2016: (Start)
G.f.: (-1 + 7*x + 12*x^2 + 82*x^3 + 17*x^4 + 3*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)
a(n) = A131471(n) - 1. - Omar E. Pol, Apr 05 2016

A189118 a(n) = smallest composite (odd) number greater than a(n-1) such that a(n)+2n is the first prime after a(n).

Original entry on oeis.org

9, 25, 91, 119, 201, 295, 527, 891, 1133, 1341, 1671, 2479, 2973, 4299, 5593, 8469, 9553, 15691, 16145, 19621, 25481, 28233, 31423, 31909, 34073, 35619, 43337, 44295, 89695, 107381, 134519, 155943, 173363, 175143, 188037, 212705, 265629

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 16 2011

nonn

More precise definition from Don Reble, Jul 29 2014. (Old definition was: Smallest composite odd number of first sequence of n consecutive composite odd numbers.)

			9 is composite but 11 is not; 25 and 27 are composite but 29 is not; 91, 93, 95 are composite but 97 is not.

Vincenzo Librandi, Table of n, a(n) for n = 1..70

Cf. A002808, A071904, A161945, A075067.

imax=1000000;lst={};q=1;Do[k=0;Do[If[PrimeQ[n],Break[],k++],{n,m,imax+200,2}];If[k==q,q++;AppendTo[lst,m]],{m,5,imax,2}];lst

Definition revised by N. J. A. Sloane, Jul 29 2014 at the suggestion of Don Reble.

A129822 Array read by rows: each row lists three consecutive nonprime odd numbers.

Original entry on oeis.org

91, 93, 95, 115, 117, 119, 117, 119, 121, 119, 121, 123, 121, 123, 125, 141, 143, 145, 143, 145, 147, 183, 185, 187, 185, 187, 189, 201, 203, 205, 203, 205, 207, 205, 207, 209, 213, 215, 217, 215, 217, 219, 217, 219, 221, 243, 245, 247, 245, 247, 249, 285

Offset: 1

Roger L. Bagula, May 20 2007

Robert Israel, Table of n, a(n) for n = 1..10002

Cf. A099019 (second column), A161945 (first column).

a:=proc(n) if isprime(2*n-1)=false and isprime(2*n+1)=false and isprime(2*n+3)=false then 2*n-1, 2*n+1, 2*n+3 else fi end: seq(a(n),n=1..150);

Flatten[Table[If[OddQ[n - 1] && PrimeQ[n - 1] == False && PrimeQ[n + 1] == False && PrimeQ[n + 3] == False, {{n - 1}, {n + 1}, {n + 3}}, {}], {n, 2, 300}]]
Select[Partition[Range[1,301,2],3,1],NoneTrue[#,PrimeQ]&]//Flatten (* Harvey P. Dale, Jul 16 2021 *)

Edited by N. J. A. Sloane, Jun 09 2007

Definition modified by Harvey P. Dale, Jul 16 2021

A133609 Numbers k such that k, k+2 and k+4 are consecutive semiprimes.

Original entry on oeis.org

183, 287, 319, 411, 413, 469, 515, 533, 579, 667, 685, 789, 813, 1055, 1077, 1133, 1145, 1165, 1203, 1253, 1313, 1347, 1383, 1385, 1387, 1389, 1561, 1685, 1687, 1793, 1795, 1817, 1839, 1849, 1919, 1937, 1957, 1959, 2045, 2047, 2155, 2227, 2315, 2317

Offset: 1

Zak Seidov, Dec 28 2007

nonn

Terms k in A136196 such that k+2 are also in A136196.

All terms are odd, so it is a subsequence of A161945. - Michel Marcus, Oct 15 2013

			183, 185 and 187 are 59th, 60th and 61st semiprimes,
287, 289 and 291 are 89th, 90th and 91st semiprimes,
319, 321 and 323 are 101st, 102nd and 103rd semiprimes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A136196.

Select[Range[2317],AllTrue[{#,#+2,#+4},PrimeOmega[#]==2&]&&AllTrue[{#+1,#+3},PrimeOmega[#]!=2&]&] (* James C. McMahon, Mar 29 2025 *)

isok(n) = (bigomega(n) == 2) && (bigomega(n+1) != 2) && (bigomega(n+2) == 2) && (bigomega(n+3) != 2) && (bigomega(n+4) == 2); \\ Michel Marcus, Oct 15 2013

A189119 Sums k of three consecutive odd numbers, all of which are composite, such that k is also the smallest in a set of three consecutive odd numbers, all of which are composite.

Original entry on oeis.org

621, 867, 891, 897, 1023, 1239, 1413, 1587, 1881, 2091, 2115, 2145, 2169, 2403, 2451, 2505, 2601, 2769, 2871, 2889, 3003, 3129, 3171, 3231, 3237, 3243, 3399, 3417, 3423, 3435, 3441, 3471, 3477, 3501, 3807, 3813, 3933, 3993, 4029

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 16 2011

nonn

			621 is a term: it is the smallest of three consecutive odd composite numbers (621 = 3^3*23, 623 = 7*89, 625 = 5^4) and is also the sum of three consecutive odd composite numbers (205 = 5*41, 207 = 3^2*23, 209 = 11*19, and 205 + 207 + 209 = 621).

Cf. A002808, A161945.

fQ[n_]:=!PrimeQ[n]&&!PrimeQ[n+2]&&!PrimeQ[n+4];
lst1=Select[Range[3,9000,2],fQ];
lst2=3*Select[Range[3,3000,2],fQ]+6;
Intersection[lst1,lst2]

Name clarified by Tanya Khovanova, Jun 23 2021

cp's OEIS Frontend

A271209 a(n) = n^5 + n + 1.

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A271208 a(n) = n^5 + n - 1.

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A189118 a(n) = smallest composite (odd) number greater than a(n-1) such that a(n)+2n is the first prime after a(n).

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A129822 Array read by rows: each row lists three consecutive nonprime odd numbers.

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A133609 Numbers k such that k, k+2 and k+4 are consecutive semiprimes.

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A189119 Sums k of three consecutive odd numbers, all of which are composite, such that k is also the smallest in a set of three consecutive odd numbers, all of which are composite.

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