A097338 -id:A097338

A173059

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

+0
21

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

MATHEMATICA

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

PROG

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

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

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

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

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Feb 08 2010

EXTENSIONS

Definition clarified by Zak Seidov, Jul 11 2014

STATUS

approved

A153045

Numbers k such that 2*k-11 is not a prime.

+0
4

10, 13, 16, 18, 19, 22, 23, 25, 28, 30, 31, 33, 34, 37, 38, 40, 43, 44, 46, 48, 49, 51, 52, 53, 55, 58, 61, 63, 64, 65, 66, 67, 68, 70, 72, 73, 76, 77, 78, 79, 82, 83, 85, 86, 88, 90, 91, 93, 94, 97, 98, 99, 100, 103, 106, 107, 108, 109, 110, 112, 113, 114

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The terms are the values of 2*h*k + k + h + 6, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

LINKS

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

FORMULA

a(n) = 5+A104275(n+1). [R. J. Mathar, Oct 22 2009]

MATHEMATICA

Select[Range[10, 200], !PrimeQ[2*#-11]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

PROG

(Magma) [n: n in [7..120] | not IsPrime(2*n - 11)]; // Vincenzo Librandi, Oct 11 2012

(Python)

from sympy import isprime

def ok(n): return n > 6 and not isprime(2*n-11)

print(list(filter(ok, range(115)))) # Michael S. Branicky, Oct 13 2021

CROSSREFS

Cf. A097338, A153043, A163652, A104275.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Dec 17 2008

STATUS

approved

A153081

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

+0
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

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

EXAMPLE

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.

MATHEMATICA

(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 *)

PROG

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

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

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

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

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Vincenzo Librandi, Dec 18 2008

EXTENSIONS

Edited and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

STATUS

approved

A153143

Nonnegative numbers k such that 2k + 19 is prime.

+0
25

0, 2, 5, 6, 9, 11, 12, 14, 17, 20, 21, 24, 26, 27, 30, 32, 35, 39, 41, 42, 44, 45, 47, 54, 56, 59, 60, 65, 66, 69, 72, 74, 77, 80, 81, 86, 87, 89, 90, 96, 102, 104, 105, 107, 110, 111, 116, 119, 122, 125, 126, 129, 131, 132, 137, 144, 146, 147, 149, 156, 159, 164, 165

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Or, (p-19)/2 for primes p >= 19.

a(n) = (A000040(n+7) - 19)/2.

a(n) = A005097(n+6) - 9.

a(n) = A067076(n+6) - 8.

a(n) = A089038(n+5) - 7.

a(n) = A105760(n+4) - 6.

a(n) = A101448(n+3) - 4.

a(n) = A089559(n+1) - 2.

LINKS

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

EXAMPLE

For k = 4, 2*k+19 = 27 is not prime, so 4 is not in the sequence;

for k = 17, 2*k+19 = 53 is prime, so 17 is in the sequence.

MATHEMATICA

(Prime[Range[8, 100]]-19)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

Select[Range[0, 170], PrimeQ[(2*#)+19]&] (* Vincenzo Librandi, Sep 24 2012 *)

PROG

(Magma) [ n: n in [0..165] | IsPrime(2*n+19) ];

(PARI) is(n)=isprime(2*n+19) \\ Charles R Greathouse IV, Feb 17 2017

(Sage) [n for n in (0..200) if is_prime(2*n+19) ] # G. C. Greubel, May 22 2019

(GAP) Filtered([0..200], k-> IsPrime(2*k+19) ) # G. C. Greubel, May 22 2019

CROSSREFS

Cf. A000040 (prime numbers), A153144 (2n+19 is not prime).

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), this seq (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).

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Dec 19 2008

EXTENSIONS

Edited, corrected and extended by Klaus Brockhaus, Dec 22 2008

Definition clarified by Zak Seidov, Jul 11 2014

STATUS

approved

A118469

Triangle read by rows: a(n,m) = If(n = 1, then 1, else Prime(n) - 1 + Sum_{k=n..m} (Prime(k + 1) - Prime(k))/2 ).

+0
0

1, 1, 3, 1, 4, 5, 1, 6, 7, 8, 1, 7, 8, 9, 11, 1, 9, 10, 11, 13, 14, 1, 10, 11, 12, 14, 15, 17, 1, 12, 13, 14, 16, 17, 19, 20, 1, 15, 16, 17, 19, 20, 22, 23, 25, 1, 16, 17, 18, 20, 21, 23, 24, 26, 29, 1, 19, 20, 21, 23, 24, 26, 27, 29, 32, 33, 1, 21, 22, 23, 25, 26, 28, 29, 31, 34, 35

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

An improved triangular Goldbach sequence in which the gap sum is taken from a start at n.

LINKS

Table of n, a(n) for n=1..77.

EXAMPLE

1

1, 3

1, 4, 5

1, 6, 7, 8

1, 7, 8, 9, 11

1, 9, 10, 11, 13, 14

1, 10, 11, 12, 14, 15, 17

1, 12, 13, 14, 16, 17, 19, 20

1, 15, 16, 17, 19, 20, 22, 23, 25

1, 16, 17, 18, 20, 21, 23, 24, 26, 29

MATHEMATICA

t[n_, m_] := If[n == 1, 1, Prime[n] + Sum[(Prime[k + 1] - Prime[k])/2, {k, n, m}] - 1]; Table[ t[n, m], {m, 11}, {n, m}] // Flatten

CROSSREFS

Main diagonal: A078444, 2nd diagonal: A073273.

Columns 1-8: A000012, A006254, A098090, A089253, A097069, A097338, A097480, A098605.

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, May 04 2006

STATUS

approved

A105760

Nonnegative numbers k such that 2k+7 is prime.

+0
32

0, 2, 3, 5, 6, 8, 11, 12, 15, 17, 18, 20, 23, 26, 27, 30, 32, 33, 36, 38, 41, 45, 47, 48, 50, 51, 53, 60, 62, 65, 66, 71, 72, 75, 78, 80, 83, 86, 87, 92, 93, 95, 96, 102, 108, 110, 111, 113, 116, 117, 122, 125, 128, 131, 132, 135, 137, 138, 143, 150, 152, 153, 155, 162

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.

EXAMPLE

If n=0, then 2*0 + 7 = 7 (prime).

If n=15, then 2*15 + 7 = 37 (prime).

If n=27, then 2*27 + 7 = 61 (prime).

MATHEMATICA

(Prime[Range[4, 100]]-7)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

Select[Range[0, 200], PrimeQ[2 # + 7] &] (* Vincenzo Librandi, May 20 2014 *)

PROG

(Magma)[n: n in [0..200]| IsPrime(2*n+7)]; // Vincenzo Librandi, Dec 21 2010

(PARI) is(n)=isprime(2*n+7) \\ Charles R Greathouse IV, Feb 16 2017

(Sage) [n for n in (0..200) if is_prime(2*n+7) ] # G. C. Greubel, May 21 2019

(GAP) Filtered([0..200], k-> IsPrime(2*k+7) ) # G. C. Greubel, May 21 2019

CROSSREFS

Cf. A153053 (Numbers n such that 2n+7 is not a prime)

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), this seq(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), A097932 (k=19).

KEYWORD

easy,nonn

AUTHOR

Parthasarathy Nambi, May 04 2005

EXTENSIONS

More terms from Rick L. Shepherd, May 18 2005

STATUS

approved

A101448

Nonnegative numbers k such that 2k + 11 is prime.

+0
23

0, 1, 3, 4, 6, 9, 10, 13, 15, 16, 18, 21, 24, 25, 28, 30, 31, 34, 36, 39, 43, 45, 46, 48, 49, 51, 58, 60, 63, 64, 69, 70, 73, 76, 78, 81, 84, 85, 90, 91, 93, 94, 100, 106, 108, 109, 111, 114, 115, 120, 123, 126, 129, 130, 133, 135, 136, 141, 148, 150, 151, 153, 160, 163

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

2 is the smallest single-digit prime and 11 is the smallest two-digit prime.

LINKS

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

EXAMPLE

If n=1, then 2*1 + 11 = 13 (prime).

If n=49, then 2*49 + 11 = 109 (prime).

If n=69, then 2*69 + 11 = 149 (prime).

MAPLE

select(k-> isprime(11+2*k), [$0..200])[]; # Alois P. Heinz, Jun 02 2022

MATHEMATICA

Select[Range[0, 200], PrimeQ[2# + 11] &] (* Stefan Steinerberger, Feb 28 2006 *)

PROG

(Magma) [n: n in [0..200] | IsPrime(2*n+11)] // Vincenzo Librandi, Nov 17 2010

(PARI) is(n)=isprime(2*n+11) \\ Charles R Greathouse IV, Apr 29 2015

(Sage) [n for n in (0..200) if is_prime(2*n+11) ] # G. C. Greubel, May 21 2019

(GAP) Filtered([0..200], k-> IsPrime(2*k+11) ) # G. C. Greubel, May 21 2019

CROSSREFS

Cf. A101123, A101086.

Numbers n such that 2n+k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), A155722 (k=9), this seq (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), A097932 (k=19).

KEYWORD

easy,nonn

AUTHOR

Parthasarathy Nambi, Jan 24 2005

EXTENSIONS

More terms from Stefan Steinerberger, Feb 28 2006

Definition clarified by Zak Seidov, Jul 11 2014

STATUS

approved

A097932

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

+0
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

MATHEMATICA

(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 *)

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

A097069

Positive integers n such that 2n - 9 is prime.

+0
22

6, 7, 8, 10, 11, 13, 14, 16, 19, 20, 23, 25, 26, 28, 31, 34, 35, 38, 40, 41, 44, 46, 49, 53, 55, 56, 58, 59, 61, 68, 70, 73, 74, 79, 80, 83, 86, 88, 91, 94, 95, 100, 101, 103, 104, 110, 116, 118, 119, 121, 124, 125, 130, 133, 136, 139, 140, 143, 145, 146, 151, 158, 160

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

Half of p+9 where p is a prime greater than 2.

MATHEMATICA

(Prime[Range[2, 100]]+9)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)

Select[Range[4, 200], PrimeQ[2 # - 9] &] (* Vincenzo Librandi, Oct 16 2012 *)

PROG

(Magma) [n: n in [6..160] | IsPrime(2*n-9)]; // Bruno Berselli, Mar 05 2011

(PARI) is(n)=isprime(2*n-9) \\ Charles R Greathouse IV, Apr 28 2015

CROSSREFS

Cf. A000040, A086801.

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), this seq(k=9), A097338 (k=11), A097363 (k=13), A097480 (k=15), A098605 (k=17), A097932 (k=19).

KEYWORD

easy,nonn

AUTHOR

Douglas Winston (douglas.winston(AT)srupc.com), Sep 15 2004

STATUS

approved

A097363

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

+0
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

FORMULA

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

MATHEMATICA

(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 *)

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved