A067257 -id:A067257 - cp's OEIS frontend

A067256 Numbers n such that n, 2n+1, 3n+2 are primes.

3, 5, 23, 29, 83, 89, 173, 233, 239, 293, 419, 659, 953, 1013, 1223, 1409, 1559, 1583, 1889, 2003, 2129, 2339, 2549, 2693, 2939, 3359, 3389, 3593, 3803, 4349, 4373, 4409, 4919, 4943, 5333, 6113, 6173, 8093, 8273, 8513, 9059, 9479, 9539, 10163, 10313

Offset: 1

Benoit Cloitre, Feb 20 2002

nonn

a(n)*(2a(n)+1)*(3a(n)+2) are Lucas-Carmichael numbers for n > 1. Analogous to A174734 as A006972 (Lucas-Carmichael numbers) is analogous to A002997 (Carmichael numbers). - Amiram Eldar, Aug 11 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A005384, A006972, A067257, A067258, A101767, A101768, A101769, A101770, A174734, A216925.

[n: n in [1..15000] | IsPrime(n) and IsPrime(2*n+1) and IsPrime(3*n+2)]; // Vincenzo Librandi, Oct 31 2014

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

is(n)=isprime(3*n+2) && isprime(2*n+1) && isprime(n) \\ Charles R Greathouse IV, Sep 14 2015

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

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)

A336059 Numbers p such that p, 2p-1, 3p-2, 4p-3 are primes.

Original entry on oeis.org

331, 1531, 3061, 4261, 4951, 6841, 10831, 15391, 18121, 23011, 25411, 26041, 31771, 33301, 40111, 41491, 45061, 49831, 53881, 59341, 65851, 70141, 73771, 78541, 88741, 95461, 96931, 109471, 111721, 112621, 117721, 131311, 133201, 134731, 135301, 150151, 165901

Offset: 1

Jeppe Stig Nielsen, Jul 07 2020

The subset p, 2p-1, 4p-3 is a Cunningham chain of the 2nd kind, cf. A057326.

Wikipedia, Cunningham chain

Cf. A005382, A057326, A067257, A237189, A336060.

Select[Range[10^5], AllTrue[{#, 2# - 1, 3# - 2, 4# - 3}, PrimeQ] &] (* Amiram Eldar, Jul 07 2020 *)

a(n) = A237189(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

A067256 Numbers n such that n, 2n+1, 3n+2 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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A033593 a(n) = (n-1)*(2*n-1)*(3*n-1)*(4*n-1).

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A336059 Numbers p such that p, 2p-1, 3p-2, 4p-3 are primes.

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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.

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