A067076 -id:A067076 - cp's OEIS frontend

A097932 Positive integers n such that 2n-19 is prime.

11, 12, 13, 15, 16, 18, 19, 21, 24, 25, 28, 30, 31, 33, 36, 39, 40, 43, 45, 46, 49, 51, 54, 58, 60, 61, 63, 64, 66, 73, 75, 78, 79, 84, 85, 88, 91, 93, 96, 99, 100, 105, 106, 108, 109, 115, 121, 123, 124, 126, 129, 130, 135, 138, 141, 144, 145, 148, 150, 151, 156, 163

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 21 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), this sequence (k=19).

(Prime[Range[2,100]]+19)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[10,200],PrimeQ[2#-19]&] (* Harvey P. Dale, May 08 2017 *)

Half of p+19 where p is a prime greater than 2.

A153081 Nonnegative numbers k such that 2k + 13 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 23, 24, 27, 29, 30, 33, 35, 38, 42, 44, 45, 47, 48, 50, 57, 59, 62, 63, 68, 69, 72, 75, 77, 80, 83, 84, 89, 90, 92, 93, 99, 105, 107, 108, 110, 113, 114, 119, 122, 125, 128, 129, 132, 134, 135, 140, 147, 149, 150, 152, 159, 162, 167

Offset: 1

Vincenzo Librandi, Dec 18 2008

Or, (p-13)/2 for primes p >= 13.
a(n) = (A000040(n+5) - 13)/2.
a(n) = A005097(n+4) - 6.
a(n) = A067076(n+4) - 5.
a(n) = A089038(n+3) - 4.
a(n) = A105760(n+2) - 3.
a(n) = A101448(n+1) - 1.
a(n) = A089559(n-1) + 1 for n > 1.

			For k = 7, 2*k+13 = 27 is not prime, so 7 is not in the sequence;
for k = 8, 2*k+13 = 29 is prime, so 8 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040 (prime numbers).
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), this seq (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+13) ); # G. C. Greubel, May 22 2019

[ n: n in [0..200] | IsPrime(2*n+13) ];

(Prime[Range[6,100]]-13)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0,200],PrimeQ[2#+13]&] (* Harvey P. Dale, Mar 02 2015 *)

is(n)=isprime(2*n+13) \\ Charles R Greathouse IV, Jul 12 2016

[n for n in (0..200) if is_prime(2*n+13) ] # G. C. Greubel, May 22 2019

Edited and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

A097338 Positive integers n such that 2n-11 is prime.

Original entry on oeis.org

7, 8, 9, 11, 12, 14, 15, 17, 20, 21, 24, 26, 27, 29, 32, 35, 36, 39, 41, 42, 45, 47, 50, 54, 56, 57, 59, 60, 62, 69, 71, 74, 75, 80, 81, 84, 87, 89, 92, 95, 96, 101, 102, 104, 105, 111, 117, 119, 120, 122, 125, 126, 131, 134, 137, 140, 141, 144, 146, 147, 152, 159, 161

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 17 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), this sequence (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+11)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[6,200],PrimeQ[2#-11]&] (* Harvey P. Dale, May 24 2014 *)

is(n)=isprime(2*n-11) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+11 where p is a prime greater than 2.

A097480 Positive integers n such that 2n-15 is prime.

Original entry on oeis.org

9, 10, 11, 13, 14, 16, 17, 19, 22, 23, 26, 28, 29, 31, 34, 37, 38, 41, 43, 44, 47, 49, 52, 56, 58, 59, 61, 62, 64, 71, 73, 76, 77, 82, 83, 86, 89, 91, 94, 97, 98, 103, 104, 106, 107, 113, 119, 121, 122, 124, 127, 128, 133, 136, 139, 142, 143, 146, 148, 149, 154, 161, 163

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 19 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602, A098603.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), this sequence (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+15)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[9,200],PrimeQ[2#-15]&] (* Harvey P. Dale, Apr 04 2021 *)

is(n)=isprime(2*n-15) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+15 where p is a prime greater than 2.

A098605 Positive integers n such that 2n - 17 is prime.

Original entry on oeis.org

10, 11, 12, 14, 15, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 38, 39, 42, 44, 45, 48, 50, 53, 57, 59, 60, 62, 63, 65, 72, 74, 77, 78, 83, 84, 87, 90, 92, 95, 98, 99, 104, 105, 107, 108, 114, 120, 122, 123, 125, 128, 129, 134, 137, 140, 143, 144, 147, 149, 150, 155, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 20 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602, A098603.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), this sequence (k=17), A097932 (k=19).

(Prime[Range[2,100]]+17)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

is(n)=isprime(2*n-17) \\ Charles R Greathouse IV, Jul 12 2016

Half of p+17 where p is a prime greater than 2.

A173059 Nonnegative numbers k such that 2*k + 17 is prime.

Original entry on oeis.org

0, 1, 3, 6, 7, 10, 12, 13, 15, 18, 21, 22, 25, 27, 28, 31, 33, 36, 40, 42, 43, 45, 46, 48, 55, 57, 60, 61, 66, 67, 70, 73, 75, 78, 81, 82, 87, 88, 90, 91, 97, 103, 105, 106, 108, 111, 112, 117, 120, 123, 126, 127, 130, 132, 133, 138, 145, 147, 148, 150, 157, 160, 165

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), this seq (k=17), A153143 (k=19).

Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

Filtered([0..200], k-> IsPrime(2*k+17) ); # G. C. Greubel, May 22 2019

[n: n in [0..200] | IsPrime(2*n+17) ]; // G. C. Greubel, May 22 2019

Mathematica
```
(Prime[Range[7,100]]-17)/2
```

is(n)=isprime(2*n+17) \\ Charles R Greathouse IV, Feb 17 2017

[n for n in (0..200) if is_prime(2*n+17) ] # G. C. Greubel, May 22 2019

Definition clarified by Zak Seidov, Jul 11 2014

A097363 Positive integers n such that 2n-13 is prime.

Original entry on oeis.org

8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 25, 27, 28, 30, 33, 36, 37, 40, 42, 43, 46, 48, 51, 55, 57, 58, 60, 61, 63, 70, 72, 75, 76, 81, 82, 85, 88, 90, 93, 96, 97, 102, 103, 105, 106, 112, 118, 120, 121, 123, 126, 127, 132, 135, 138, 141, 142, 145, 147, 148, 153, 160, 162

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 18 2004

Shawn A. Broyles, Table of n, a(n) for n = 1..1000

Cf. A000040, A098602.
Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).
Numbers n such that 2n-k is prime: A006254 (k=1), A098090 (k=3), A089253 (k=5), A089192 (k=7), A097069 (k=9), A097338 (k=11), this sequence (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

(Prime[Range[2,100]]+13)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[8,200],PrimeQ[2#-13]&] (* Harvey P. Dale, Apr 26 2013 *)

Half of p+13 where p is a prime greater than 2.

A054269 Length of period of continued fraction for sqrt(prime(n)).

Original entry on oeis.org

1, 2, 1, 4, 2, 5, 1, 6, 4, 5, 8, 1, 3, 10, 4, 5, 6, 11, 10, 8, 7, 4, 2, 5, 11, 1, 12, 6, 15, 9, 12, 6, 9, 18, 9, 20, 17, 18, 4, 5, 14, 21, 16, 13, 1, 20, 26, 4, 2, 5, 11, 12, 17, 14, 1, 12, 3, 24, 21, 13, 18, 5, 14, 16, 17, 11, 34, 19, 14, 7, 15, 4, 20, 5, 30, 8, 9, 21, 1, 21, 18, 37, 16

Offset: 1

N. J. A. Sloane, May 05 2000

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004
Note that primes of the form n^2+1 (A002496) have a continued fraction whose period length is 1; odd primes of the form n^2+2 (A056899) have length 2; odd primes of the form n^2-2 (A028871) have length 4. - T. D. Noe, Nov 03 2006
For an odd prime p, the length of the period is odd if p=1 (mod 4) or even if p=3 (mod 4). - T. D. Noe, May 22 2007

T. D. Noe, Table of n, a(n) for n = 1..10000
A. I. Gliga, On continued fractions of the square root of prime numbers

Cf. A003285, A130272 (primes at which the period length sets a new record).

with(numtheory): for i from 1 to 150 do cfr := cfrac(ithprime(i)^(1/2), 'periodic','quotients'); printf(`%d,`, nops(cfr[2])) od:

Table[p=Prime[n]; Length[Last[ContinuedFraction[Sqrt[p]]]],{n,100}] (* T. D. Noe, May 22 2007 *)
Length[ContinuedFraction[Sqrt[#]][[2]]]&/@Prime[Range[100]] (* Harvey P. Dale, Sep 28 2024 *)

More terms from James Sellers, May 05 2000

A086801 a(n) = prime(n) - 3.

Original entry on oeis.org

-1, 0, 2, 4, 8, 10, 14, 16, 20, 26, 28, 34, 38, 40, 44, 50, 56, 58, 64, 68, 70, 76, 80, 86, 94, 98, 100, 104, 106, 110, 124, 128, 134, 136, 146, 148, 154, 160, 164, 170, 176, 178, 188, 190, 194, 196, 208, 220, 224, 226, 230, 236, 238, 248, 254, 260, 266, 268, 274, 278

Offset: 1

Giovanni Teofilatto, Aug 05 2003

Numbers n such that n+3 is a prime.

			n=16 is here because 16+3=19 is a prime.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

Cf. A006254, A067076, A091738, A116127, A119981, A173064.

Prime[Range[22]]-3 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

for(x=2,20,print1(prime(x)-3,",")) \\ Jorge Coveiro, Jan 30 2006

a(n) = 2*A067076(n) = 2*(A006254(n+1)-2).

Extended by Ray Chandler, Nov 29 2003

A153238 Numbers k such that 2*k + 3 is composite.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 16, 18, 21, 23, 24, 26, 27, 30, 31, 33, 36, 37, 39, 41, 42, 44, 45, 46, 48, 51, 54, 56, 57, 58, 59, 60, 61, 63, 65, 66, 69, 70, 71, 72, 75, 76, 78, 79, 81, 83, 84, 86, 87, 90, 91, 92, 93, 96, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 111, 114

Offset: 1

Vincenzo Librandi, Dec 21 2008

One less than the associated value in A047845. - R. J. Mathar, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Marc Wolf, François Wolf, Representation theorem of composite odd numbers indices, SCIREA Journal of Mathematics (2018) 3.3, 106-117.

Cf. A067076, A047845, A144562.

[ n: n in [0..120] | not IsPrime(2*n+3) ]; // Klaus Brockhaus, Dec 19 2010

lst={};Do[If[ !PrimeQ[2*n+3],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Select[Range[120], CompositeQ[2 # + 3] &] (* Michael De Vlieger, Oct 17 2018 *)

for(n=1,115,if(!isprime(2*n+3),print1(n,", "))) \\ Joerg Arndt, Dec 19 2010

a(n) ~ n. - Charles R Greathouse IV, Dec 28 2011

Edited by Ray Chandler, Jan 07 2009

cp's OEIS Frontend

A097932 Positive integers n such that 2n-19 is prime.

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A153081 Nonnegative numbers k such that 2k + 13 is prime.

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A097338 Positive integers n such that 2n-11 is prime.

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A097480 Positive integers n such that 2n-15 is prime.

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A098605 Positive integers n such that 2n - 17 is prime.

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A173059 Nonnegative numbers k such that 2*k + 17 is prime.

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Magma

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Sage

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A097363 Positive integers n such that 2n-13 is prime.

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A054269 Length of period of continued fraction for sqrt(prime(n)).

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