A067076 -id:A067076 - cp's OEIS frontend

A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

284, 3074, 3494, 21698, 32138, 43874, 51794, 60674, 75494, 407348, 437438, 459794, 571478, 660878, 667358, 705464, 716624, 740774, 811028, 820154, 910664, 1059398, 1077998, 1122584, 1150748, 1210754, 1222898, 1265018, 1412174, 1461164, 1486574, 1585868, 1631438

Offset: 1

Zak Seidov, Jun 03 2005

nonn

n == 0 (mod 2). n == 2 (mod 3). n == 3 or 4 (mod 5). - Jason Yuen, Sep 02 2024

			284 is OK because 2*284+3, 3*284+5, 5*284+7, 7*284+11, 11*284+13 and 13*284+17 all are primes.

Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5).

Cf. A108117 (k=1..7), A379427 (k=1..8).

s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 6}]]=={True}, s=Append[s, n]], {n, 2, 1000000, 2}];s

\\ See isok from A108117
for(n=1,2*10^6,if(isok(n,6),print1(n", "))) \\ Jason Yuen, Sep 02 2024

a(22)-a(33) from Jason Yuen, Sep 02 2024

A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.

Original entry on oeis.org

3494, 60674, 75494, 1122584, 2136044, 2473934, 3367244, 5600384, 6629804, 6910784, 7554644, 8572904, 10079144, 11848094, 11892164, 12043214, 12167594, 12269234, 12507284, 12700154, 13459664, 13924544, 14495354, 15005954, 16890914, 17827094, 20642984, 25796054

Offset: 1

Zak Seidov, Jun 03 2005

nonn

The only n, for which also 19*3494+23 is prime, is n=5600384. In principle, n == 4 (mod 10) can give primes of the form prime(k)*n+prime(k+1), for all k from 1 up to 41, while prime(42)*4+prime(43)=181*4+191 == 5 (mod 10) that is nonprime. It'd be very interesting to find at least one n such that prime(k)*n+prime(k+1), k=1,...,41 are all prime.

There are no values of n such that prime(k)*n+prime(k+1), k=1,...,9 are all prime. Proof: If n = 3*i then 2*(3*i)+3 = 3*(2*i+1) is not prime. If n = 3*i+1 then 5*(3*i+1)+7 = 3*(5*i+4) is not prime. If n = 3*i+2 then 23*(3*i+2)+29 = 3*(23*i+25) is not prime. - Jason Yuen, Sep 02 2024

			3494 is OK because 2*3494+3, 3*3494+5, 5*3494+7, 7*3494+11, 11*3494+13, 13*3494+17 and 17*3494+19 all are primes.

Cf. A067076 (k=1), A088879 (k=2), A111224 (k=3), A101123 (k=4), A102721 (k=5), A108976 (k=7).

Cf. A108110 (k=1..6), A379427 (k=1..8).

s={};Do[If[Union[PrimeQ/@Table[Prime[k]*n+Prime[k+1], {k, 7}]]=={True}, s=Append[s, n]], {n, 4, 10000000, 10}];s
Select[Range[9*10^6],AllTrue[Prime[Range[7]]#+Prime[Range[2,8]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 24 2018 *)

isok(n,upto=7)=for(k=1,upto,if(!isprime(prime(k)*n+prime(k+1)),return(0)));1
for(n=1,3*10^7,if(isok(n),print1(n", "))) \\ Jason Yuen, Sep 02 2024

a(13)-a(28) from Jason Yuen, Sep 02 2024

A125954 Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 11, 2, 5, 11, 2, 2, 5, 71, 2, 3, 2, 2, 167, 2, 17, 3, 2, 197, 149, 2, 2, 3, 3, 2, 2267, 2, 2, 3, 3, 2, 29, 2, 2531, 167, 2, 7, 3, 3, 2, 61, 2, 2, 11, 2, 2, 157, 2, 5, 7, 7, 149, 3, 5, 2, 379, 2, 41, 3, 2, 2, 3, 79, 11, 3, 2, 2, 97, 3, 2, 3, 3, 2, 1321, 2, 17, 31, 2, 61

Offset: 0

Alexander Adamchuk, Feb 07 2007

All terms are primes.
a(n) = 2 for n = {1,2,4,5,7,8,10,13,14,17,19,20,22,...} = A067076 Numbers n such that 2n+3 is a prime.
a(34),...,a(40) = {2,2,3,3,2,29,2}.
a(42),...,a(80) = {167,2,7,3,3,2,61,2,2,11,2,2,157,2,5,7,7,149,3,5,2,379,2,41,3,2,2,3,79,11,3,2,2,97,3,2,3,3,2}.
a(82),...,a(90) = {2,17,31,2,61,7,2,2,5}.
a(93),...,a(95) = {383,2,2}.
a(97),...,a(100) = {2,2,5,7}.
a(102),...,a(124) = {13,11,2,5,5,17,3,103,2,19,2,2,3,2,31,37,2,2,3,3,7,3,2}.
a(127),...,a(131) = {2,61,31,2,157}.
a(133),...,a(142) = {2,2,7,3,2,13,2,2,7,3}.
a(144),...,a(146) = {173,2,11}.
a(148),...,a(150) = {3,17,107}.
a(n) is currently unknown for n = {33,41,81,91,92,96,101,125,126,132,143,147,...}.

Cf. A067076.
Cf. A000043 = Primes p such that 2^p - 1 is prime.
Cf. A001348 = Mersenne numbers: 2^p - 1, where p is prime.
Cf. A057468 = numbers n such that 3^n - 2^n is prime.
Cf. A125958 = Least number k > 0 such that (2^k + (2n-1)^k)/(2n+1) is prime.

Do[k = 1; While[ !PrimeQ[((2n+1)^k - 2^k)/(2n-1)], k++ ]; Print[k], {n, 100}] (* Ryan Propper, Mar 29 2007 *)
lnk[n_]:=Module[{k=1},While[!PrimeQ[((2n+1)^k-2^k)/(2n-1)],k++];k]; Array[ lnk,90] (* Harvey P. Dale, May 19 2012 *)

More terms from Ryan Propper, Mar 29 2007

A152843 Numbers n such that both 2n+3 and 4n+7 are prime.

Original entry on oeis.org

0, 1, 4, 10, 13, 19, 25, 40, 43, 55, 64, 85, 88, 94, 115, 118, 124, 139, 145, 178, 208, 214, 220, 244, 253, 295, 319, 325, 328, 340, 358, 370, 379, 403, 454, 475, 505, 508, 514, 523, 550, 610, 613, 643, 703, 718, 724, 739, 748, 754, 778, 790, 799, 865, 904, 943

Offset: 1

Vincenzo Librandi, Dec 14 2008

Or, numbers n such that 2n+3 is a Sophie Germain prime. [Klaus Brockhaus, Dec 22 2008]

			For n = 10, 2*n+3 = 23 is prime and 4*n+7 = 47 is prime. 23 = A005384(5).

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

Cf. A067076 (2n+3 is prime), A089986 (4n+7 is prime), A005384 (Sophie Germain primes).

[ n: n in [0..1000] | IsPrime(2*n+3) and IsPrime(4*n+7) ];

Join[{0}, Select[Range[1000], PrimeQ[2*#+3] && PrimeQ[4*#+7] &]] (* Vincenzo Librandi, Aug 30 2012 *)
Select[Range[0,1000],AllTrue[{2#+3,4#+7},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 07 2015 *)

Edited and extended by Klaus Brockhaus, Dec 22 2008

A154115 Numbers n such that n + 3 is prime.

Original entry on oeis.org

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

Vladimir Joseph Stephan Orlovsky, Jan 04 2009

			a(2) = 2 since (2 + 2)^2 - (2 + 1)^2 - 2 = 5.

G. C. Greubel, Table of n, a(n) for n = 1..2000

Cf. A140719, A005097, A154111, A154112, A154113.

Cf. A067076 (a(n-1)/2).

[n: n in [0..500] | IsPrime((n+2)^2-(n+1)^2-n)]; // Vincenzo Librandi, Nov 26 2010

A154115 := proc(n) ithprime(n+1)-3 ; end proc: # R. J. Mathar, May 09 2010

a[n_]:=(n+2)^2-(n+1)^2-n;lst={};Do[If[PrimeQ[a[n]],AppendTo[lst,n]],{n,6!}];lst
Select[Range[0,300],PrimeQ[(#+2)^2-(#+1)^2-#]&] (* Harvey P. Dale, Nov 06 2013 *)
Prime[Range[2,100]]-3 (* Harvey P. Dale, Jul 15 2017 *)

is(n)=isprime(n+3) \\ Charles R Greathouse IV, Sep 02 2016

a(n) = A086801(n+1). - R. J. Mathar, May 09 2010

New name based on a comment by Franklin T. Adams-Watters, Jan 30 2009

A154591 a(n) = 2n^2 + 18n + 7.

Original entry on oeis.org

27, 51, 79, 111, 147, 187, 231, 279, 331, 387, 447, 511, 579, 651, 727, 807, 891, 979, 1071, 1167, 1267, 1371, 1479, 1591, 1707, 1827, 1951, 2079, 2211, 2347, 2487, 2631, 2779, 2931, 3087, 3247, 3411, 3579, 3751, 3927, 4107, 4291, 4479, 4671, 4867, 5067, 5271

Offset: 1

Vincenzo Librandi, Jan 12 2009

Ninth diagonal of A144562.

2*a(n) + 67 is a square.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A067076, A144562, A153238.

I:=[27, 51, 79]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012

LinearRecurrence[{3, -3, 1}, {27, 51, 79}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

for(n=1, 40, print1(2*n^2 + 18*n + 7", ")); \\ Vincenzo Librandi, Feb 22 2012

[2*n^2+18*n+7 for n in range(1,51)] #  G. C. Greubel, May 27 2024

G.f.: (9*x^2-6*x-7)/(x-1)^3. - Bruno Berselli, Dec 07 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
Sum_{n>=1} 1/a(n) = 1621/20097 + tan(sqrt(67)*Pi/2)*Pi/(2*sqrt(67)). - Amiram Eldar, Feb 25 2023
E.g.f.: (7 + 20*x + 2*x^2)*exp(x). - G. C. Greubel, May 27 2024

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Eric Desbiaux, Feb 11 2010

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 1 + 3 OR 3* 1 + 2 =  5;
  2* 4 + 3 OR 3* 3 + 2 = 11;
  2* 7 + 3 OR 3* 5 + 2 = 17;
  2*10 + 3 OR 3* 7 + 2 = 23;
  2*13 + 3 OR 3* 9 + 2 = 29;
  2*19 + 3 OR 3*13 + 2 = 41;
  2*22 + 3 OR 3*15 + 2 = 47;
  2*25 + 3 OR 3*17 + 2 = 53;
  2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Cf. A067076, A024893, A007528, A024898, A059325.

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Data corrected and extended by Amiram Eldar, Jul 30 2024

A243811 Numbers k such that 2k+3 and 2k+5 are both prime.

Original entry on oeis.org

0, 1, 4, 7, 13, 19, 28, 34, 49, 52, 67, 73, 88, 94, 97, 112, 118, 133, 139, 154, 172, 208, 214, 229, 259, 283, 298, 307, 319, 328, 403, 409, 412, 427, 439, 508, 514, 523, 529, 544, 574, 613, 637, 643, 649, 658, 712, 724, 739, 742, 802, 808, 832, 847, 859, 892, 934

Offset: 1

Vincenzo Librandi, Jun 11 2014

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

Cf. A040040, A067076, A089038.

[n: n in [0..1000] | IsPrime(2*n+3) and IsPrime(2*n+5)];

Select[Range[0, 1000], PrimeQ[2 # + 3] && PrimeQ[2 # + 5] &]

a(n) = A040040(n)-2.

A290839 a(n) = smallest prime p such that 2p + 2n - 1 is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 2, 2, 7, 3, 2, 3, 2, 2, 3, 2, 7, 3, 2, 5, 3, 2, 2, 7, 3, 2, 3, 2, 2, 13, 3, 2, 3, 2, 11, 3, 2, 5, 7, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 13, 7, 11, 5, 19, 3, 2, 3, 2, 5, 3, 2, 2, 7, 5, 5, 3, 2, 2, 7, 3, 2, 13, 3, 2, 3, 2, 7, 3, 2

Offset: 0

XU Pingya, Aug 12 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005384, A023204, A023205, A023206, A023207, A171517, A020483, A290838.

Cf. A067076 (indices n at which a(n) = 2).

Table[j=0; found=False; While[!found, j++; found=PrimeQ[2Prime[j]+2n-1]]; Prime[j], {n, 85}]

a(n) = {my(p=2); while(!isprime(2*p+2*n-1), p = nextprime(p+1)); p;} \\ Michel Marcus, Aug 12 2017

a(-n) = A290838(n+1). - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A153167 Numbers n such that n+2 is not a Chen prime.

Original entry on oeis.org

2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 88, 89, 90

Offset: 1

Vincenzo Librandi, Dec 20 2008

nonn

Contains all strictly positive even numbers A005843.
For each odd k>1 we can accumulate the numbers == k^2-2 (mod 2k) in a row, the last entry equal to A073577(k):
7; (k=3)
13, 23; (k=5)
19, 33, 47; (k=7)
25, 43, 61, 79; (k=9)
31, 53, 75, 97, 119; (k=11)
7, 63, 89, 115, 141, 167; (k=13)
43, 73, 103, 133, 163, 193,223; (k=17)
49, 83, 17, 151,185, 219, 253, 287; (k=19)
Each element T of this table has the format T= k^2-2-j*2*k, so T+2 is of the form k*(k-2*j), therefore not prime, and consequently all elements T are in the sequence.

Cf. A109611, A040976, A073577, A067076.

Edited, 41, 59 (see A102540) etc. inserted by R. J. Mathar, Oct 16 2009

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A108110 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,6 all are primes.

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A108117 Numbers n such that prime(k)*n+prime(k+1), for k=1,...,7 all are primes.

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A125954 Least number k > 0 such that ((2n+1)^k - 2^k)/(2n-1) is prime.

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A152843 Numbers n such that both 2n+3 and 4n+7 are prime.

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Magma

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A154115 Numbers n such that n + 3 is prime.

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A154591 a(n) = 2*n^2 + 18*n + 7.

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A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

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A154591 a(n) = 2n^2 + 18n + 7.

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

A243811 Numbers k such that 2k+3 and 2k+5 are both prime.