A067256 -id:A067256 - cp's OEIS frontend

A067257 Numbers n such that n, 2n+1, 3n+2, 4n+3 are primes.

5, 89, 1409, 1889, 10589, 11549, 11909, 12899, 17159, 19889, 22349, 24239, 26189, 35999, 37049, 37379, 39419, 44879, 45569, 49919, 60779, 67559, 68669, 75329, 83579, 88919, 104369, 108359, 112349, 114599, 127139, 133979, 135029, 135449

Offset: 1

Benoit Cloitre, Feb 20 2002

nonn

Terms are congruent to {5,29} mod 30. - Zak Seidov, May 31 2012

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000040, A005384, A067256, A067258, A101767-A101770.

[n: n in [1..150000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(3*n+2) and IsPrime(4*n+3)]; // Vincenzo Librandi, Oct 31 2014

Select[Prime[Range[10^5]],PrimeQ[2*#+1]&&PrimeQ[3*#+2]&&PrimeQ[4*#+3] &] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

Extended by Ray Chandler, Dec 31 2004

A067258 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4 are primes.

Original entry on oeis.org

5, 89, 12899, 35999, 45569, 83579, 108359, 154769, 175349, 196769, 206009, 209039, 303029, 374009, 420419, 489179, 513239, 641549, 658349, 709589, 765749, 775949, 862769, 991079, 1018709, 1057019, 1265549, 1527629, 1609739, 1621079

Offset: 1

Benoit Cloitre, Feb 20 2002

Except for 5, all terms == 29 (mod 30). - Robert Israel, May 28 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A005384, A067256, A067257, A101767-A101770.

select(t -> andmap(isprime, [t,2*t+1,3*t+2,4*t+3,5*t+4]),
[5, seq(i,i=29..2*10^6,30)]); # Robert Israel, May 28 2018

a={};Do[p=Prime[n];If[PrimeQ[p*2+1]&&PrimeQ[p*3+2]&&PrimeQ[p*4+3]&&PrimeQ[p*5+4],AppendTo[a,p]],{n,1,10^5}];Print[a]; (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

More terms from Sascha Kurz, Mar 23 2002

A101767 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4, 6n+5 are primes.

Original entry on oeis.org

154769, 175349, 641549, 658349, 1018709, 2274089, 2894219, 5246009, 6621929, 7949759, 8189999, 8678669, 10366439, 12327629, 13951559, 18160379, 18924569, 21914339, 22279949, 22297799, 24765509, 25592279, 31029389, 31835159, 36802079, 38844119, 38911949

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

a(n) == 209 (mod 210) - John Cerkan, Mar 22 2018

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

Cf. A000040, A005384, A067256, A067257, A067258, A101768, A101769, A101770.

a={}; Do[p=Prime[n]; If[PrimeQ[p*2+1]&&PrimeQ[p*3+2]&&PrimeQ[p*4+3]&&PrimeQ[p*5+4]&&PrimeQ[p*6+5], AppendTo[a, p]], {n, 1, 10^5}]; Print[a]; (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

Terms a(25) and beyond from John Cerkan, Mar 22 2018

A101770 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4, 6n+5, 7n+6, 8n+7, 9n+8 are primes.

Original entry on oeis.org

407874179, 1674689729, 6380217479, 15002412599, 24291715139, 28081637219, 34274541839, 37048322849, 45785202539, 53434060679, 100061694809, 101245430999, 103024911989, 127890675989, 130173995279, 141481942139, 149397940019, 177352532069, 212815427999, 214580145779, 294249502259, 296754699779

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

nonn

All terms == 2099 or 2309 (mod 2310). - Robert Israel, Jul 05 2016

Robert Israel, Table of n, a(n) for n = 1..100

Cf. A000040, A005384, A067256, A067257, A067258, A101767, A101768, A101769.

select(n -> andmap(isprime,
[n,2*n+1,3*n+2,4*n+3,5*n+4,6*n+5,7*n+6,8*n+7,9*n+8]),
[seq(seq(2310*i+j, j=[2099,2309]),i=0..10^7)]); # Robert Israel, Jul 05 2016

More terms from Jens Kruse Andersen, May 08 2008

A101769 Numbers p such that p, 2p+1, 3p+2, 4p+3, 5p+4, 6p+5, 7p+6, 8p+7 are primes.

Original entry on oeis.org

2894219, 60041519, 64523969, 242024369, 407874179, 1092040949, 1092075389, 1674689729, 2281060319, 5035134509, 5329406669, 5683382879, 5792424329, 6000216809, 6380217479, 10409580719, 11488703939, 13745865209, 14181824369, 14904963149, 15002412599, 15412603919

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

From Jeppe Stig Nielsen, Jul 07 2020: (Start)
Each term is -1 modulo 210.
The subset p, 2p+1, 4p+3, 8p+7 is a Cunningham chain, cf. A023272. (End)

Robert Israel, Table of n, a(n) for n = 1..250 (first 50 terms from Jeppe Stig Nielsen)
Wikipedia, Cunningham chain

Cf. A000040, A005384, A023272, A067256, A067257, A067258, A101767, A101768, A101769, A101770, A336060.

R:= NULL: count:= 0:
for i from 0 while count < 50 do
  for j in [1049,2099, 2309] do
    p:= 2310*i+j;
    if andmap(isprime,[p, 2*p + 1, 3*p + 2, 4*p + 3, 5*p + 4, 6*p + 5, 7*p + 6, 8*p + 7]) then
      count:= count+1; R:= R,p;
    fi
od od:
R; # Robert Israel, May 21 2025

a={}; Do[p=Prime[n]; If[PrimeQ[p*2+1]&&PrimeQ[p*3+2]&&PrimeQ[p*4+3]&&PrimeQ[p*5+4]&&PrimeQ[p*6+5]&&PrimeQ[p*7+6]&&PrimeQ[p*8+7], AppendTo[a, p]], {n, 1, 10^7}]; Print[a]; (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(20)-a(22) from Jeppe Stig Nielsen, Jul 07 2020

A101768 Numbers k such that k, 2k+1, 3k+2, 4k+3, 5k+4, 6k+5, 7k+6 are primes.

Original entry on oeis.org

2894219, 5246009, 8189999, 8678669, 13951559, 18160379, 24765509, 38911949, 40645919, 60041519, 64523969, 108405989, 124028309, 126392699, 132767039, 142738679, 189142589, 242024369, 248451839, 325561319, 354218759, 392136359

Offset: 1

Jonathan Vos Post and Ray Chandler, Dec 31 2004

Cf. A000040, A005384, A067256-A067258, A101767-A101770.

a={}; Do[p=Prime[n]; If[PrimeQ[p*2+1] && PrimeQ[p*3+2] && PrimeQ[p*4+3] && PrimeQ[p*5+4] && PrimeQ[p*6+5] && PrimeQ[p*7+6], AppendTo[a, p]], {n, 1, 17^5}]; a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Range[4*10^8], And@@PrimeQ[#*Range[7]+Range[0,6]]&] (* Harvey P. Dale, Jul 22 2011 *)

A033594 a(n) = (n-1)(2n-1)(3n-1).

Original entry on oeis.org

-1, 0, 15, 80, 231, 504, 935, 1560, 2415, 3536, 4959, 6720, 8855, 11400, 14391, 17864, 21855, 26400, 31535, 37296, 43719, 50840, 58695, 67320, 76751, 87024, 98175, 110240, 123255, 137256, 152279, 168360

Offset: 0

N. J. A. Sloane

The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384 and the subsequence of those for which in addition (3*n+2) is prime is A067256. - Jonathan Vos Post, Dec 15 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005384, A016921, A067256.

List([0..40], n-> (n-1)*(2*n-1)*(3*n-1) ); # G. C. Greubel, Mar 05 2020

[(n-1)*(2*n-1)*(3*n-1): n in [0..40]]; // Vincenzo Librandi, May 24 2011

A033594:=n->(n-1)*(2*n-1)*(3*n-1); seq(A033594(n), n=0..40); # Wesley Ivan Hurt, Feb 24 2014

Table[(n-1)*(2*n-1)*(3*n-1),{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{4,-6,4,-1},{-1,0,15,80},40] (* Harvey P. Dale, Aug 23 2012 *)

vector(41, n, my(m=n-1); (m-1)*(2*m-1)*(3*m-1) ) \\ G. C. Greubel, Mar 05 2020

[-1]+[n^3*rising_factorial((n-1)/n, 3) for n in (1..40)] # G. C. Greubel, Mar 05 2020

a(n)*A016921(n) + 1 = A051866(n)^2. - Bruno Berselli, May 23 2011
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=-1, a(1)=0, a(2)=15, a(3)=80. - Harvey P. Dale, Aug 23 2012
G.f.: (-1 +4*x +9*x^2 +24*x^3)/(1-x)^4. - R. J. Mathar, Feb 06 2017
E.g.f.: (-1 + x + 7*x^2 + 6*x^3)*exp(x). - G. C. Greubel, Mar 05 2020
From Amiram Eldar, Jan 03 2021: (Start)
Sum_{n>=2} 1/a(n) = (7 - sqrt(3)*Pi - 16*log(2) + 9*log(3))/4.
Sum_{n>=2} (-1)^n/a(n) = Pi - 7/4 - sqrt(3)*Pi/2 + 2*log(2). (End)

A033593 a(n) = (n-1)(2n-1)(3n-1)(4n-1).

Original entry on oeis.org

1, 0, 105, 880, 3465, 9576, 21505, 42120, 74865, 123760, 193401, 288960, 416185, 581400, 791505, 1053976, 1376865, 1768800, 2238985, 2797200, 3453801, 4219720, 5106465, 6126120, 7291345, 8615376, 10112025, 11795680, 13681305, 15784440, 18121201, 20708280, 23562945

Offset: 0

N. J. A. Sloane

The sequence of n such that n is prime and (2*n+1) is prime is the sequence of Sophie Germain primes A005384; the subsequence of those for which in addition (3*n+2) is prime is A067256; and the subsequence of those for which in addition (4*n+3) is prime is A067257. - Jonathan Vos Post, Dec 15 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n) = A011245(-n).

Cf. A005384, A033594, A067256, A067257.

[ 24*n^4-50*n^3+35*n^2-10*n+1: n in [0..40]]; // Vincenzo Librandi, Jan 30 2011

[&*[s*n-1: s in [1..4]]: n in [0..40]]; // Bruno Berselli, May 23 2011

1, seq( n^4*pochhammer((n-1)/n, 4), n=1..40); # G. C. Greubel, Mar 05 2020

Table[1-10 n+35 n^2-50 n^3+24 n^4,{n,0,40}] (* or *) LinearRecurrence[{5,-10, 10,-5,1}, {1,0,105,880,3465}, 40]  (* Harvey P. Dale, Jan 29 2011 & Apr 26 2011 *)

a(n)=24*n^4-50*n^3+35*n^2-10*n+1 \\ Charles R Greathouse IV, May 23 2011

[1]+[n^4*rising_factorial((n-1)/n, 4) for n in (1..40)] # G. C. Greubel, Mar 05 2020

G.f.: (1 -5*x +115*x^2 +345*x^3 +120*x^4)/(1-x)^5. - R. J. Mathar, Jan 30 2011
From G. C. Greubel, Mar 05 2020: (Start)
a(n) = n^4* Pochhammer((n-1)/n, 4).
E.g.f.: (1 - x + 53*x^2 + 94*x^3 + 24*x^4)*exp(x). (End)
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=2} 1/a(n) = 29/36 + (4/3 - 3*sqrt(3)/4)*Pi - 12*log(2) +  27*log(3)/4.
Sum_{n>=2} (-1)^n/a(n) = (1 + 4*sqrt(2)/3 - 3*sqrt(3)/2)*Pi + 14*log(2)/3 - 4*sqrt(2)*log(2)/3 + 8*sqrt(2)*log(2-sqrt(2))/3 - 29/36. (End)

A290810 Numbers k such that 6k-1, 12k-1 and 18k-1 are all primes.

Original entry on oeis.org

1, 4, 5, 14, 15, 29, 39, 40, 49, 70, 110, 159, 169, 204, 235, 260, 264, 315, 334, 355, 390, 425, 449, 490, 560, 565, 599, 634, 725, 729, 735, 820, 824, 889, 1019, 1029, 1349, 1379, 1419, 1510, 1580, 1590, 1694, 1719, 1765, 1925, 1930, 1950, 1985, 2044, 2150

Offset: 1

Amiram Eldar, Aug 11 2017

nonn

If k is in the sequence then (6k-1)(12k-1)(18k-1) = 36k * (36k^2 - 11k + 1) - 1 is a Lucas-Carmichael number (A006972).

Analogous to A046025 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers).

			1 is in the sequence since 6*1 - 1 = 5, 12*1 - 1 = 11 and 18*1 - 1 = 17 are all primes, and 5*11*17 = 935 is a Lucas-Carmichael number.

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

Cf. A002997, A006972, A046025, A067256, A087788, A216925.

Intersection of A024898, A138620 and A138918.

seq = {}; Do[ If[ AllTrue[{6 m - 1, 12 m - 1, 18 m - 1}, PrimeQ ], AppendTo[seq, m] ], {m, 1, 10^5} ]; seq

isok(n) = isprime(6*n-1) && isprime(12*n-1) && isprime(18*n-1); \\ Michel Marcus, Aug 11 2017

6*a(n) - 1 = A067256(n+1).

A101779 a(n) = least k such that all of k, 2k+1, 3k+2, ..., nk+n-1 are primes, or 0 if no such k is found.

Original entry on oeis.org

2, 2, 3, 5, 5, 154769, 2894219, 2894219, 407874179, 214580145779, 9448481062019, 247236503934419, 2545206711847799, 18178612369988250179, 53792264108455702829

Offset: 1

Jonathan Vos Post and Ray Chandler, Jan 13 2005

nonn

a(10) > 3691000000, Robert G. Wilson v, Mar 23 2007

By definition the same as A088651(n)-1 if k exists. It is conjectured k always exists. - a(10)-a(15) from Jens Kruse Andersen, May 02 2008

Cf. A000040, A005384, A067256, A067257, A067258, A101767, A101768, A101769, A101770.

Cf. A088651.

f[1] = 2; f[n_] := f[n] = Block[{k = PrimePi@ f[n - 1], p, t = Table[i*p + (i - 1), {i, 2, n}]}, While[p = Prime@k; Union@PrimeQ@t != {True}, k++ ]; p]; Do[ Print[f@n // Timing], {n, 10}] (* Robert G. Wilson v, Mar 23 2007 *)

a(10)-a(15) from Jens Kruse Andersen, May 02 2008

cp's OEIS Frontend

A067257 Numbers n such that n, 2n+1, 3n+2, 4n+3 are primes.

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A067258 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4 are primes.

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A101767 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4, 6n+5 are primes.

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A101770 Numbers n such that n, 2n+1, 3n+2, 4n+3, 5n+4, 6n+5, 7n+6, 8n+7, 9n+8 are primes.

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A101769 Numbers p such that p, 2p+1, 3p+2, 4p+3, 5p+4, 6p+5, 7p+6, 8p+7 are primes.

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A101768 Numbers k such that k, 2*k+1, 3*k+2, 4*k+3, 5*k+4, 6*k+5, 7*k+6 are primes.

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A033594 a(n) = (n-1)*(2*n-1)*(3*n-1).

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A033593 a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).

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A101768 Numbers k such that k, 2k+1, 3k+2, 4k+3, 5k+4, 6k+5, 7k+6 are primes.

A033594 a(n) = (n-1)(2n-1)(3n-1).

A033593 a(n) = (n-1)(2n-1)(3n-1)(4n-1).