A095278 -id:A095278 - cp's OEIS frontend

A023281 Primes that remain prime through 3 iterations of function f(x) = 4x + 3.

2, 109, 179, 571, 677, 977, 1279, 1447, 1747, 1901, 2207, 2671, 3119, 3917, 5011, 5399, 5441, 5569, 5791, 6211, 6607, 7079, 7417, 8369, 8831, 9221, 9697, 9769, 11821, 11897, 12347, 13537, 13669, 13691, 13729, 13781, 13907, 14747, 14851, 15581, 17231, 17497

Offset: 1

David W. Wilson

nonn

Primes p such that 4*p+3, 16*p+15 and 64*p+63 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023213, A023250, and of A095278.

[n: n in [1..150000] | IsPrime(n) and IsPrime(4*n+3) and IsPrime(16*n+15) and IsPrime(64*n+63)] // Vincenzo Librandi, Aug 04 2010

f:=proc(x) options operator, arrow: 4*x+3 end proc: a:=proc(n) if isprime(n)= true and isprime(f(n))=true and isprime(f(f(n)))=true and isprime(f(f(f(n)))) =true then n else end if end proc: seq(a(n),n=1..20000); # Emeric Deutsch, Jan 01 2008

Select[Prime@ Range@ 2100, Times @@ Boole@ PrimeQ@ Rest@ NestList[4 # + 3 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)

is(n)=isprime(n) && isprime(4*n+3) && isprime(16*n+15) && isprime(64*n+63) \\ Charles R Greathouse IV, Sep 20 2016

A105135 Numbers n such that 32n+17 is prime.

Original entry on oeis.org

0, 3, 7, 10, 12, 13, 18, 27, 30, 31, 37, 40, 42, 46, 48, 55, 58, 66, 67, 75, 81, 87, 88, 90, 96, 97, 100, 103, 115, 117, 118, 121, 126, 130, 132, 133, 135, 142, 145, 147, 150, 156, 163, 165, 168, 172, 195, 198, 201, 202, 205, 208, 210, 213, 217, 220, 222, 235, 243, 250, 252

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105133-A105140, A002145, A007521, A105126-A105132.

[n: n in [0..500] | IsPrime(32*n+17)]; // Vincenzo Librandi, Jan 07 2013

Select[Range[0, 260], PrimeQ[32 # + 17]&] (* Harvey P. Dale, Apr 26 2011 *)

is(n)=isprime(32*n+17) \\ Charles R Greathouse IV, Jun 12 2017

A105136 Numbers n such that 64n+33 is prime.

Original entry on oeis.org

1, 5, 10, 14, 19, 26, 29, 31, 32, 35, 40, 41, 47, 49, 52, 56, 62, 64, 70, 80, 82, 91, 95, 104, 115, 116, 119, 122, 127, 130, 134, 136, 139, 146, 151, 160, 161, 164, 166, 179, 181, 182, 196, 197, 206, 211, 214, 217, 221, 224, 227, 230, 235, 236, 239, 244, 250, 251, 256, 257

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105133-A105140, A002145, A007521, A105126-A105132.

Select[Range[300],PrimeQ[64#+33]&] (* Harvey P. Dale, Sep 19 2011 *)

is(n)=isprime(64*n+33) \\ Charles R Greathouse IV, Jun 12 2017

A105137 Numbers n such that 128n+65 is prime.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 21, 24, 33, 36, 37, 42, 43, 46, 49, 54, 58, 61, 66, 67, 72, 79, 81, 88, 93, 94, 102, 103, 106, 112, 114, 123, 124, 126, 138, 148, 154, 157, 163, 166, 168, 177, 186, 187, 196, 198, 199, 201, 204, 207, 211, 213, 214, 219, 231, 232, 238, 252, 256, 262, 264

Offset: 1

N. J. A. Sloane, based on correspondence from Marco Matosic, Apr 11 2005

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

Cf. A095278, A105133-A105140, A002145, A007521, A105126-A105132.

Subsequence of A032766.

[n: n in [0..500] | IsPrime(128*n+65)]; // Vincenzo Librandi, Jan 07 2013

Select[Range[500], PrimeQ[128 # + 65]&] (* Vincenzo Librandi, Jan 07 2013 *)

is(n)=isprime(128*n+65) \\ Charles R Greathouse IV, Jun 12 2017

A124855 Numbers k such that 3k + 4 and 4k + 3 are primes.

Original entry on oeis.org

1, 5, 11, 19, 25, 31, 41, 49, 59, 65, 89, 91, 109, 115, 121, 125, 151, 161, 179, 181, 205, 209, 229, 241, 245, 275, 305, 329, 331, 349, 355, 361, 371, 389, 415, 439, 509, 515, 521, 535, 551, 595, 599, 625, 661, 665, 671, 719, 725, 749, 755, 769, 779, 791, 839

Offset: 1

Zak Seidov, Nov 10 2006

nonn

Intersection of A034936 and A095278. Prime terms are in A106068.

[n: n in [0..1000] | IsPrime(3*n+4) and IsPrime(4*n+3)] // Vincenzo Librandi, Mar 26 2010

Select[Range[850], PrimeQ[3# + 4] && PrimeQ[4# + 3] &] (* Ray Chandler, May 11 2007 *)

A244087 Numbers n such that 4n+3 and 8n+7 are prime.

Original entry on oeis.org

0, 2, 5, 20, 32, 44, 47, 59, 62, 89, 104, 107, 110, 122, 164, 170, 179, 185, 227, 254, 257, 275, 305, 359, 362, 374, 377, 389, 395, 452, 482, 500, 509, 515, 584, 587, 599, 614, 635, 674, 704, 725, 734, 740, 755, 824, 839, 872, 884, 905, 944, 950, 962, 965, 977

Offset: 1

Vincenzo Librandi, Jun 25 2014

-2 is a primitive root mod (8*n+7).

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

Intersection of A095278 and A139487.

[n: n in [0..1000] | IsPrime(4*n+3) and IsPrime(8*n+7)];

Select[Range[0, 1500], PrimeQ[4 # + 3]&&PrimeQ[8 # + 7] &]

A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

Original entry on oeis.org

1, 370, 41425, 81535, 255625, 267175, 311590, 365350, 1054570, 1381750, 2533600, 2975125, 3266080, 3930205, 4684210, 4782385, 4802860, 5940850, 6414610, 7986565, 8429245, 8570470, 8636305, 8810080, 9270715, 9857980, 10459525, 13708225, 13917490, 15127720, 15252460

Offset: 1

K. D. Bajpai, Jan 27 2015

nonn

All terms in this sequence are congruent to 1 mod 3.

Subsequence of A123986.

			a(2) = 370;
4*370 +  1 = 1481;
4*370 +  3 = 1483;
4*370 +  7 = 1487;
4*370 +  9 = 1489;
4*370 + 13 = 1493;
4*370 + 19 = 1499;
All six are prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 726 terms from K. D. Bajpai)

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^8] | forall{4*n+i: i in [1, 3, 7, 9, 13, 19] |  IsPrime(4*n+i)}]; // Vincenzo Librandi, Mar 12 2015

Select[Range[5*10^7], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] && PrimeQ[4*# + 19] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13, 19} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13) &&isprime(4*n + 19) , print1(n,", ")))

A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

Original entry on oeis.org

1, 25, 370, 4015, 4855, 10945, 36040, 41425, 41710, 50455, 56335, 61900, 81535, 86995, 116290, 129700, 134110, 158365, 207430, 239635, 255625, 265990, 267175, 272815, 293395, 311590, 335080, 337810, 339700, 342115, 365350, 393385, 403960, 481345, 488590, 550990

Offset: 1

K. D. Bajpai, Jan 29 2015

nonn

All terms in this sequence are 1 mod 3.
Each term yields a set of five consecutive primes.
Alternatively: Numbers n such that 4n+k forms a set of five consecutive primes for k = {1,3,7,9,13}.
Subsequence of A123986.

			25 is in the list because 4*25 + 1 = 101, 4*25 + 3 = 103, 4*25 + 7 = 107, 4*25 + 9 = 109 and 4*25 + 13 = 113 are all prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8458

Cf. A000040, A005098, A095278, A123986.

[n: n in [0..10^6] | forall{4*n+r: r in [1,3,7,9,13] | IsPrime(4*n+r)}]; // Vincenzo Librandi, Feb 16 2015

Select[Range[1, 500000], PrimeQ[4*# + 1] && PrimeQ[4*# + 3] && PrimeQ[4*# + 7] && PrimeQ[4*# + 9] && PrimeQ[4*# + 13] &]
Select[Range[5*10^6], And @@ PrimeQ /@ ({1, 3, 7, 9, 13} + 4 #) &]

for(n=1,10^7, if( isprime(4*n + 1) && isprime(4*n + 3) &&isprime(4*n + 7) &&isprime(4*n + 9) &&isprime(4*n + 13), print1(n,", ")))

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.

Original entry on oeis.org

8, 11, 13, 16, 22, 26, 28, 29, 31, 35, 37, 38, 41, 43, 44, 50, 53, 56, 59, 64, 65, 68, 70, 73, 74, 76, 80, 85, 86, 88, 91, 97, 98, 107, 109, 112, 113, 116, 118, 121, 122, 125, 127, 133, 134, 136, 137, 139, 142, 145, 146, 149, 151, 152, 155, 160, 161, 167, 170

Offset: 1

K. D. Bajpai, Aug 29 2020

nonn

Integers that are in A067076 or in A095278, but not in both. - Michel Marcus, Aug 29 2020

			a(1) = 8 is a term because 2*8 + 3 = 19 is a prime; but 4*8 + 3 = 35 = (5*7) is a composite number.
a(4) = 16 is a term because 2*16 + 3 = 35 = (5*7) is a composite number; but 4*16 + 3 = 67  is a prime.
a(6) = 26 is a term because 2*26 + 3 = 55 = (5*11) is a composite number; but 4*26 + 3 = 107  is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5000

Cf. A063908, A067076, A095278, A337480.

A337491:=n->`if`((isprime(2*n+3) xor isprime(4*n+3)), n, NULL): seq(A337491(n), n=1..500);

Select[Range[0, 250], Xor[PrimeQ[2 # + 3], PrimeQ[4 # + 3]] &]
Select[Range[200],Total[Boole[PrimeQ[{2,4}#+3]]]==1&] (* Harvey P. Dale, Jan 26 2023 *)

isok(k) = bitxor(isprime(2*k+3), isprime(4*k+3)); \\ Michel Marcus, Aug 29 2020

cp's OEIS Frontend

A023281 Primes that remain prime through 3 iterations of function f(x) = 4x + 3.

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Magma

Maple

Mathematica

PARI

A105135 Numbers n such that 32n+17 is prime.

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Magma

Mathematica

PARI

A105136 Numbers n such that 64n+33 is prime.

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Mathematica

PARI

A105137 Numbers n such that 128n+65 is prime.

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Magma

Mathematica

PARI

A124855 Numbers k such that 3k + 4 and 4k + 3 are primes.

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Magma

Mathematica

A244087 Numbers n such that 4*n+3 and 8*n+7 are prime.

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Magma

Mathematica

A254288 Numbers k such that 4*k + {1, 3, 7, 9, 13, 19} are all prime.

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Magma

Mathematica

PARI

A254376 Numbers n such that 4n+1, 4n+3, 4n+7, 4n+9 and 4n+13 are prime.

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A244087 Numbers n such that 4n+3 and 8n+7 are prime.

A337491 Numbers k such that exactly one of 2k + 3 and 4k + 3 is prime.