A024899 -id:A024899 - cp's OEIS frontend

A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.

6, 11, 13, 16, 21, 26, 27, 35, 37, 46, 51, 55, 56, 61, 62, 63, 66, 68, 73, 76, 81, 83, 90, 91, 96, 101, 102, 105, 112, 115, 118, 121, 122, 123, 125, 126, 128, 131, 142, 146, 151, 153, 156, 161, 165, 166, 168, 173, 178, 181, 186, 187, 188, 195, 200, 202, 206, 208, 216

Offset: 1

Lekraj Beedassy, Aug 20 2006

nonn

Entries in A046953 which are not in A060461 or equivalently, entries in A024899 which are not in A002822.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A121764.

Filtered([1..250], k-> not IsPrime(6*k-1) and IsPrime(6*k+1)); # G. C. Greubel, Feb 20 2019

[n: n in [1..250] | not IsPrime(6*n-1) and  IsPrime(6*n+1)]; // G. C. Greubel, Feb 20 2019

Select[Range[250], ! PrimeQ[6# - 1] && PrimeQ[6# + 1] &] (* Ray Chandler, Aug 22 2006 *)

for(n=1, 250, if(!isprime(6*n-1) && isprime(6*n+1), print1(n", "))) \\ G. C. Greubel, Feb 20 2019

[n for n in (1..250) if not is_prime(6*n-1) and  is_prime(6*n+1)] # G. C. Greubel, Feb 20 2019

Extended by Ray Chandler, Aug 22 2006

A153218 Numbers k such that 6k + 7 is prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17, 20, 22, 24, 25, 26, 29, 31, 32, 34, 36, 37, 39, 44, 45, 46, 50, 51, 54, 55, 57, 60, 61, 62, 65, 67, 69, 71, 72, 75, 76, 80, 82, 86, 89, 90, 94, 95, 99, 100, 101, 102, 104, 106, 109, 111, 114, 117, 120, 121, 122, 124, 125, 127, 130

Offset: 1

Vincenzo Librandi, Dec 21 2008

One less than the associated term in A024899. - R. J. Mathar, Jan 05 2011

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

Cf. A153219.

[n: n in [0..150] | IsPrime(6*n + 7)]; // Vincenzo Librandi, Sep 24 2012

Select[Range[0, 150], PrimeQ[6# + 7] &] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(6*n+7) \\ Charles R Greathouse IV, Sep 24 2012

A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

Original entry on oeis.org

1, 3, 18, 105, 135, 153, 165, 168, 300, 363, 585, 618, 648, 765, 828, 1110, 1140, 1278, 1518, 1530, 1533, 2130, 2223, 2400, 2475, 2613, 2790, 2925, 3075, 3180, 3345, 3420, 3483, 3810, 3840, 3843, 3933, 4008, 4083, 4095, 4143, 4260, 4263, 4323, 4470, 4545

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A005097, A005098, A024899, A123998, A124409, A124410, A124411, A071576.

Select[Range[4600], And @@ PrimeQ /@ ({2, 4, 6}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 3, isprime(2*j*k+1)) == 3; \\ Jinyuan Wang, Aug 04 2019

A124409 Numbers k such that 2k+1, 4k+1, 6k+1 and 8k+1 are primes.

Original entry on oeis.org

165, 765, 1530, 2130, 2475, 3420, 5415, 7695, 9060, 11505, 12705, 13020, 15885, 16650, 20055, 20745, 22530, 24915, 26940, 29670, 32925, 35070, 36885, 39270, 44370, 47730, 48465, 54735, 55860, 56310, 58860, 65655, 66600, 67365, 67650

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..2000

Cf. A005097, A005098, A024899, A005123, A123998, A124408, A124410, A124411, A071576.

Select[Range[68000], And @@ PrimeQ /@ ({2, 4, 6, 8}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 4, isprime(2*j*k+1)) == 4; \\ Jinyuan Wang, Aug 04 2019

A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

Original entry on oeis.org

5415, 12705, 13020, 44370, 82950, 98280, 105525, 112200, 115140, 123855, 134250, 134460, 187740, 188745, 210165, 225705, 247170, 256410, 296310, 302085, 367875, 375645, 382890, 399585, 404040, 476340, 487830, 526845, 532095, 566430, 578085

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A005097, A005098, A024899, A005123, A024912, A123998, A124408, A124409, A124411, A071576.

Select[Range[600000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 5, isprime(2*j*k+1)) == 5; \\ Jinyuan Wang, Aug 04 2019

A124411 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1, 10k+1 and 12k+1 are primes.

Original entry on oeis.org

12705, 13020, 105525, 256410, 966840, 1707510, 1944495, 2310000, 2478630, 3132675, 3836070, 3976770, 4112430, 4532325, 5499585, 5920005, 6610485, 7390845, 8552250, 10739505, 11120340, 12231450, 12338130, 13243230, 16467255

Offset: 1

Artur Jasinski, Oct 31 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..100

Cf. A005097, A005098, A024899, A005123, A024912, A110801, A123998, A124408, A124409, A124410, A071576.

Select[Range[10^7], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12}*# + 1) &] (* Ray Chandler, Nov 20 2006 *)

is(k) = sum(j = 1, 6, isprime(2*j*k+1)) == 6; \\ Jinyuan Wang, Aug 04 2019

Extended by Ray Chandler, Nov 20 2006

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341

Offset: 1

David W. Wilson

nonn

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.

Entries in chains are congruent to {1,7,3,9} mod 10.

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766.

Subsequence of A007693, A023256, and A024899.

[n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010

k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Additional comments from Labos Elemer, Jul 23 2003

A167056 Numbers k such that 12*k + 7 is prime.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 16, 17, 18, 22, 23, 25, 27, 30, 31, 36, 38, 40, 41, 43, 45, 47, 50, 51, 52, 53, 57, 60, 61, 62, 65, 67, 68, 71, 73, 75, 76, 80, 82, 86, 87, 88, 90, 93, 97, 102, 106, 107, 108, 110, 116, 118, 120, 121, 122, 123, 127, 128, 130, 131, 135, 138

Offset: 1

Michael B. Porter, Oct 27 2009

Corresponds to odd numbers in A024899.

			2 is in the sequence since 12*2+7 = 31 is prime.

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

A110801, A167055, A167057, A024899, primes are in A068229.

[n: n in [1..150] | IsPrime(12*n+7)]; // Vincenzo Librandi, May 20 2014

Select[Range[0, 200], PrimeQ[12 # + 7] &] (* Vincenzo Librandi, May 20 2014 *)

PARI
```
isA167056(n) = isprime(12*n+7)
```

A089953 Numbers n such that 3*n+7 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 18, 20, 22, 24, 30, 32, 34, 40, 44, 48, 50, 52, 58, 62, 64, 68, 72, 74, 78, 88, 90, 92, 100, 102, 108, 110, 114, 120, 122, 124, 130, 134, 138, 142, 144, 150, 152, 160, 164, 172, 178, 180, 188, 190, 198, 200, 202, 204, 208, 212, 218, 222, 228

Offset: 1

Giovanni Teofilatto, Jan 12 2004

nonn

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.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A002476 gives primes, A034936, A024892, A024899.

is(n)=isprime(3*n+7) \\ Charles R Greathouse IV, May 22 2017

a(n) = A034936(n)-1 = A024892(n)-2 = 2*A024899(n)-2.

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A167021 a(n) = 1 iff 6n+1 is prime.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1

Offset: 1

Washington Bomfim, Oct 27 2009

nonn

			a(3) = 1 because 6*3+1 is prime;
a(4) = 0 since 6*4+1 is composite.

Index entries for characteristic functions

Cf. A024899, A167020, A132350. For n < 14, a(n) = A132350(n).

[IsPrime(6*n+1) select 1 else 0:n in [1..105]]; // Marius A. Burtea, Oct 06 2019

If[PrimeQ[6#+1],1,0]&/@Range[120] (* Harvey P. Dale, Apr 03 2012 *)

a(n) = isprime(6*n+1); \\ Michel Marcus, Jan 19 2019

cp's OEIS Frontend

A121765 Numbers n such that 6*n-1 is composite while 6*n+1 is prime.

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A153218 Numbers k such that 6k + 7 is prime.

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A124408 Numbers k such that 2k+1, 4k+1 and 6k+1 are primes.

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A124409 Numbers k such that 2k+1, 4k+1, 6k+1 and 8k+1 are primes.

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A124410 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1 and 10k+1 are primes.

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A124411 Numbers k such that 2k+1, 4k+1, 6k+1, 8k+1, 10k+1 and 12k+1 are primes.

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A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

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A167056 Numbers k such that 12*k + 7 is prime.

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A121765 Numbers n such that 6n-1 is composite while 6n+1 is prime.