A005123 -id:A005123 - cp's OEIS frontend

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

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

A153762 Numbers k such that 8k + 9 is prime.

Original entry on oeis.org

1, 4, 8, 10, 11, 13, 16, 23, 28, 29, 31, 34, 38, 41, 43, 49, 50, 53, 55, 56, 64, 70, 71, 73, 74, 76, 79, 83, 94, 95, 100, 106, 109, 115, 116, 118, 121, 125, 128, 130, 136, 140, 143, 148, 149, 151, 155, 160, 161, 164, 169, 175, 178, 184, 185, 193, 199, 200, 206, 211

Offset: 1

Vincenzo Librandi, Jan 01 2009

One less than the associated term in A005123. - R. J. Mathar, Jan 05 2011

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

Cf. A023232.

[n: n in [0..300] | IsPrime(8*n + 9)]; // Vincenzo Librandi, Sep 24 2012

Select[Range[0, 250],PrimeQ[8*# + 9]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(8*n+9) \\ Charles R Greathouse IV, Sep 24 2012

A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

Original entry on oeis.org

9, 11, 13, 19, 25, 15, 21, 23, 35, 41, 25, 21, 37, 49, 23, 39, 29, 25, 57, 35, 27, 59, 65, 33, 43, 29, 49, 55, 51, 41, 37, 69, 81, 39, 59, 35, 65, 71, 77, 83, 51, 67, 47, 43, 79, 39, 97, 69, 49, 59, 41, 87, 93, 61, 47, 57, 89, 53, 101, 79, 59, 85, 55, 91, 103, 81, 115, 53, 49, 63, 83, 73, 111, 59

Offset: 1

Wolfdieter Lang, Feb 18 2015

The corresponding term y = y2(n) of this fundamental solution of the second class of the (generalized) Pell equation x^2 - 2*y^2 = -A007519(n) = -(1 + 8*A005123(n)) is given in A254937(n).

For comments and the Nagell reference see A254934.

			The first pairs [x2(n), y2(n)] of the fundamental positive solutions of this second class are (the prime A007519(n) appears as first entry):
  [17, [9, 7]], [41, [11, 9]], [73, [13, 11]],
  [89, [19, 15]], [97, [25, 19]], [113, [15, 13]],
  [137, [21, 17]], [193, [23, 19]], [233, [35, 27]],
  [241, [41, 31]], [257, [25, 21]], [281, [21, 19]],
  [313, [37, 29]], [337, [49, 37]], [353, [23, 21]],
  [401, [39, 31]], [409, [29, 25]], [433, [25, 23]],
  [449, [57, 43]], [457, [35, 29]], [521, [27, 25]],
  [569, [59, 45]], [577, [65, 49]], [593, [33, 29]],
  [601, [43, 35]], [617, [29, 27]], [641, [49, 39]], ...
a(4) = -(3*3 - 4*7) = 28 - 9 = 19.

M. F. Hasler, Table of n, a(n) for n = 1..1000, May 22 2025

Cf. A007519 (primes == 1 mod 8), A005123 (8k+1 is prime).

Cf. A254937 (corresponding y2-values), A254934 (x1 values), A254935 (y1 values), A255233 (same for primes == 7 mod 8), A255247.

apply( {A254936(n, p=A007519(n))=n=Set(abs(qfbsolve(Qfb(-1, 0, 2), p, 1)))[1]*[-3,4]~}, [1..77]) \\ The 2nd optional arg allows to directly specify the prime. - M. F. Hasler, May 22 2025

a(n)^2 - 2*A254937(n)^2 = -A007519(n) gives the second smallest positive (proper) solution of this (generalized) Pell equation.

a(n) = -(3*A254934(n) - 4*A254935(n)), n >= 1.

More terms from M. F. Hasler, May 22 2025

A230076 a(n) = (A007521(n)-1)/4.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 25, 27, 37, 39, 43, 45, 49, 57, 67, 69, 73, 79, 87, 93, 97, 99, 105, 115, 127, 135, 139, 153, 163, 165, 169, 175, 177, 183, 189, 193, 199, 205, 207, 213, 219, 235, 249, 253, 255, 265, 267, 273, 277, 279, 295, 303, 307

Offset: 1

Wolfdieter Lang, Oct 24 2013

Because A007521(n) are the primes congruent 5 (mod 8) it is clear that a(n) is congruent 1 (mod 2), that is odd.

2*a(n) = A055034(A007521(n)), the degree of the minimal polynomial C(A007521(n), x) of 2*rho(Pi/A007521(n)) (see A187360).

			The minimal polynomial C(A007521(2), x) = C(13, x) has degree 6 = 2*a(2) because C(13, x) = x^6 - x^5 - 5*x^4 + 4*x^3 + 6*x^2 - 3*x -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007521, A055034, A187360, 4*A005123 (1 (mod 8) case), A186287 (3 (mod 8) case), A186302 (7 (mod 8) case).

(Select[8*Range[0, 200] + 5, PrimeQ] - 1)/4 (* Amiram Eldar, Jun 08 2022 *)

a(n) = (A007521(n)-1)/4.

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407

Offset: 1

David W. Wilson

nonn

			First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003

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

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.

Subsequence of A005123, A023228, A023260, A023291, and A023319.

k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)

{p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.

a(n) == 197 (mod 210). - John Cerkan, Nov 04 2016

A023291 Primes that remain prime through 3 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

2, 1487, 2207, 2927, 8807, 11117, 16187, 17657, 26357, 44927, 45377, 48497, 91757, 110237, 117167, 122327, 125387, 126107, 145007, 170927, 174527, 190787, 193847, 203897, 230567, 244247, 246017, 270287, 280547, 283937, 347957, 362237, 364337

Offset: 1

David W. Wilson

nonn

Primes p such that 8*p+1, 64*p+9 and 512*p+73 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A005123, A023228, and A023260.

[n: n in [1..450000] | IsPrime(n) and IsPrime(8*n+1) and IsPrime(64*n+9) and IsPrime(512*n+73)]; // Vincenzo Librandi, Aug 04 2010

okQ[n_]:=And@@PrimeQ[NestList[8#+1&,n,3]]; Select[Prime[Range[50000]],okQ] (* Harvey P. Dale, Jan 09 2011 *)

a(n) == 17 (mod 30) for n > 1. - John Cerkan, Sep 23 2016

A115248 Values such that A115247(a(n))=1. Values such that A001935(a(n))==2 (mod 4).

Original entry on oeis.org

2, 5, 9, 11, 12, 14, 17, 19, 24, 29, 30, 32, 35, 39, 42, 44, 46, 50, 51, 53, 54, 56, 57, 65, 71, 72, 74, 75, 77, 80, 82, 84, 95, 96, 100, 101, 104, 107, 109, 110, 116, 117, 119, 122, 126, 127, 128, 129, 131, 137, 141, 144, 149, 150, 152, 154, 156, 161, 162, 165, 170

Offset: 0

Christian G. Bower, Jan 17 2006

nonn

All values are the sum of a triangular number A000217 and a square A000290.

K. Alladi, Partition Identities Involving Gaps and Weights, Transactions of the American Mathematical Society, Vol. 349, No. 12, Dec 1997, pp. 5001-5019.

Cf. A005123, A115249.

A186293 (A007519(n)-1)/2.

Original entry on oeis.org

8, 20, 36, 44, 48, 56, 68, 96, 116, 120, 128, 140, 156, 168, 176, 200, 204, 216, 224, 228, 260, 284, 288, 296, 300, 308, 320, 336, 380, 384, 404, 428, 440, 464, 468, 476, 488, 504, 516, 524, 548, 564, 576, 596, 600, 608

Offset: 1

Marco Matosic, Feb 17 2011

Cf. A186305.

(Select[1 + 8 Range@ 155, PrimeQ] -1)/2 (* Robert G. Wilson v *)

a(n) = A186294(n)-1.

a(n) = 4*A005123(n).

cp's OEIS Frontend

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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A153762 Numbers k such that 8k + 9 is prime.

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A254936 Fundamental positive solution x = x2(n) of the second class of the Pell equation x^2 - 2*y^2 = -A007519(n), n >= 1 (primes congruent to 1 mod 8).

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A230076 a(n) = (A007521(n)-1)/4.

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A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

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A023291 Primes that remain prime through 3 iterations of function f(x) = 8x + 1.

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