A129484 -id:A129484 - cp's OEIS frontend

A140444 Primes congruent to 1 (mod 14).

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x)    (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x)   (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x)   (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018

[p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018

Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)

is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Simpler definition from N. J. A. Sloane, Jul 11 2008

A140542 Primes of form 17*n - 6.

Original entry on oeis.org

11, 79, 113, 181, 283, 317, 419, 487, 521, 691, 827, 929, 997, 1031, 1201, 1303, 1439, 1609, 1847, 1949, 2017, 2153, 2221, 2357, 2459, 2663, 2731, 2833, 2969, 3037, 3343, 3547, 3581, 3853, 3989, 4057, 4091, 4159, 4261, 4363, 4397, 4567, 4703, 5009, 5077

Offset: 1

Juri-Stepan Gerasimov, Jun 28 2008

nonn

			If n=1, then 17*1-6=11. If n=7, then 17*7-6=119.

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

Cf. A129484.

Select[Range[11,5000,17],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011*)

More terms from N. J. A. Sloane, Jul 11 2008

A140545 Primes of form 17n + 6.

Original entry on oeis.org

23, 193, 227, 397, 431, 499, 601, 839, 907, 941, 1009, 1213, 1451, 1553, 1621, 1723, 2029, 2063, 2131, 2267, 2437, 2539, 2777, 2879, 3049, 3083, 3253, 3389, 3457, 3491, 3559, 3593, 3797, 3967, 4001, 4273, 4409, 4783, 4817, 4919, 4987, 5021, 5531, 5701, 5939

Offset: 1

Juri-Stepan Gerasimov, Jun 29 2008

nonn

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

Cf. A129484.

Select[ Range[6,5000,17], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011*)

forprime(p=2,1e4,if(p%17==6,print1(p", "))) \\ Charles R Greathouse IV, Dec 21 2011

More terms from N. J. A. Sloane, Jul 11 2008

A092074 Primes congruent to 3 mod 17.

Original entry on oeis.org

3, 37, 71, 139, 173, 241, 479, 547, 683, 751, 853, 887, 1091, 1193, 1499, 1567, 1601, 1669, 1873, 1907, 2111, 2179, 2213, 2281, 2383, 2417, 2621, 2689, 2791, 2927, 3301, 3539, 3607, 3709, 3947, 4049, 4219, 4253, 4423, 4457, 4729, 4831, 4933, 4967, 5171

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 29 2004

Used in a primality test.

Primes congruent to 3 mod 34. - Chai Wah Wu, Apr 29 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Chris Caldwell, Prime test.

Cf. A129484.

[ p: p in PrimesUpTo(10000) | p mod 17 eq 3 ]; // Vincenzo Librandi, Apr 08 2011

Select[ Range[3,5000,17], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011 *)
Select[Prime[Range[700]],Mod[#,17]==3&] (* Harvey P. Dale, Dec 03 2021 *)

forprime(i=1,6000,if(Mod(i,17)==3,print1(i,",")))

More terms from Mohammed Bouayoun (bouyao(AT)wanadoo.fr) and Ray Chandler, Mar 30 2004

A138631 Primes of the form 17*k + 9.

Original entry on oeis.org

43, 179, 281, 349, 383, 587, 757, 859, 1063, 1097, 1301, 1471, 1607, 1709, 1777, 1811, 1879, 1913, 2083, 2287, 2389, 2423, 2593, 2729, 2797, 3001, 3137, 3307, 3511, 3613, 3851, 3919, 4021, 4157, 4259, 4327, 4463, 4871, 4973, 5279, 5347, 5381, 5449, 5483

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

nonn

			17*2 + 9 = 43, 17*10 + 9 = 179, 17*16 + 9 = 281, 17*20 + 9 = 349, 17*22 + 9 = 349, ...

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

Cf. A138632.

Primes congruent to k mod 17: A129484 (k=1), A140544 (k=2), A092074 (k=3), A094657 (k=4), A138623 (k=5), A140545 (k=6), A138629 (k=7), A138633 (k=8), this sequence (k=9), A138627 (k=10), A140542 (k=11), A138625 (k=12), A141865 (k=13), A140540 (k=14), A140543 (k=15), A140541 (k=16).

a={};Do[x=17*n+9;If[PrimeQ[x],AppendTo[a,x]],{n,10^2}];a
Select[17*Range[350]+9,PrimeQ] (* Harvey P. Dale, May 14 2017 *)

From A.H.M. Smeets, Sep 05 2019: (Start)
a(n)/log(a(n)) ~ 16*n;
Integral_{x=2..a(n)} dx/log(x) ~ 16*n. (End)

More terms from N. J. A. Sloane, Jul 11 2008

A141865 Primes congruent to 13 mod 17.

Original entry on oeis.org

13, 47, 149, 251, 353, 421, 523, 557, 659, 727, 761, 829, 863, 1033, 1237, 1373, 1543, 1747, 1951, 2053, 2087, 2393, 2699, 2767, 2801, 2903, 2971, 3209, 3413, 3583, 3617, 3719, 3821, 3889, 3923, 4093, 4127, 4229, 4297, 4603, 4637, 4909, 4943, 5011, 5113

Offset: 1

N. J. A. Sloane, Jul 11 2008

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

Cf. A129484.

[p: p in PrimesUpTo(6000) | p mod 17 eq 13 ]; // Vincenzo Librandi, Aug 15 2012

Select[Range[13,5000,17],PrimeQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2011*)
Select[Prime[Range[700]],Mod[#,17]==13&] (* Harvey P. Dale, Jan 16 2022 *)

is(n)=isprime(n) && n%17==13 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 16n log n. - Charles R Greathouse IV, Jul 02 2016

A138633 Primes of the form 17*k - 9.

Original entry on oeis.org

59, 127, 229, 263, 331, 433, 467, 569, 739, 773, 977, 1181, 1249, 1283, 1453, 1487, 1657, 1759, 1861, 1997, 2099, 2269, 2371, 2473, 2609, 2677, 2711, 3119, 3187, 3221, 3323, 3391, 3527, 3697, 3833, 4003, 4139, 4241, 4513, 4547, 4649, 4751, 5023, 5227, 5261

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 14 2008

nonn

			17*4 - 9 = 59, 17*8 - 9 = 127, 17*14 - 9 = 229, 17*16 - 9 = 263, 17*20 - 9 = 331, 17*26 - 9 = 433, 17*28 - 9 = 467, ...

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

Cf. A138634.

Primes congruent to k mod 17: A129484 (k=1), A140544 (k=2), A092074 (k=3), A094657 (k=4), A138623 (k=5), A140545 (k=6), A138629 (k=7), this sequence (k=8), A138631 (k=9), A138627 (k=10), A140542 (k=11), A138625 (k=12), A141865 (k=13), A140540 (k=14), A140543 (k=15), A140541 (k=16).

a={};Do[x=17*n-9;If[PrimeQ[x],AppendTo[a,x]],{n,10^2}];a
Select[17*Range[400]-9,PrimeQ] (* Harvey P. Dale, Jul 25 2020 *)

From A.H.M. Smeets, Sep 05 2019: (Start)
n ~ (1/16) * a(n)/log(a(n)).
n ~ (1/16) * Integral_{x=2..a(n)} dx/log(x). (End)

More terms from N. J. A. Sloane, Jul 11 2008

A215137 a(n) = 17*n + 1.

Original entry on oeis.org

1, 18, 35, 52, 69, 86, 103, 120, 137, 154, 171, 188, 205, 222, 239, 256, 273, 290, 307, 324, 341, 358, 375, 392, 409, 426, 443, 460, 477, 494, 511, 528, 545, 562, 579, 596, 613, 630, 647, 664, 681, 698, 715, 732, 749, 766, 783, 800, 817, 834, 851, 868, 885, 902, 919, 936, 953, 970

Offset: 0

Jeremy Gardiner, Aug 04 2012

Jeremy Gardiner, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A129484, A158057, A161705.

I:=[1,18]; [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Range[1, 100, 17]
LinearRecurrence[{2,-1}, {1,18}, 50] (* G. C. Greubel, Apr 19 2018 *)

for(n=0, 50, print1(17*n + 1, ", ")) \\ G. C. Greubel, Apr 19 2018

From G. C. Greubel, Apr 19 2018: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: (1+16*x)/(1-x)^2.
E.g.f.: (17*x + 1)*exp(x). (End)

A124127 Numbers k such that 17k + 1 is prime.

Original entry on oeis.org

6, 8, 14, 18, 24, 26, 36, 38, 54, 56, 60, 66, 74, 78, 80, 84, 90, 98, 110, 116, 126, 138, 140, 150, 158, 164, 168, 180, 186, 194, 204, 210, 216, 228, 230, 236, 248, 260, 266, 270, 290, 294, 300, 308, 318, 320, 344, 356, 360, 368, 374, 378, 384, 386, 396, 404

Offset: 1

Parthasarathy Nambi, Nov 29 2006

			If k=116 then 17*k + 1 = 1973 (prime).

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

Cf. A024905, A129484 (resulting primes).

[n: n in [0..500] | IsPrime(17*n + 1)] // Vincenzo Librandi, Mar 26 2010

Select[Range[500],PrimeQ[17#+1]&] (* Harvey P. Dale, Nov 28 2019 *)

is(n)=isprime(17*n+1) \\ Charles R Greathouse IV, Jun 13 2017

A129480 a(n) = Prime(17*n).

Original entry on oeis.org

59, 139, 233, 337, 439, 557, 653, 769, 883, 1013, 1117, 1249, 1381, 1493, 1613, 1747, 1879, 2017, 2141, 2287, 2399, 2551, 2689, 2801, 2953, 3089, 3253, 3373, 3529, 3643, 3793, 3923, 4073, 4219, 4357, 4513, 4651, 4799, 4957, 5087, 5237, 5413, 5527, 5683

Offset: 1

Cino Hilliard, May 29 2007

			The 17th prime is 59.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. similar sequences listed in A031336.

Cf. A129484.

[NthPrime(17*n): n in [1..100]]; // G. C. Greubel, Feb 12 2024

seq(ithprime(17*i),i=1..100); # Robert Israel, Sep 08 2014

Prime[17*Range[1, 100]] (* G. C. Greubel, Feb 12 2024 *)

cicada(n) = forstep(x=17,n,17,print1(prime2(x)","))

a(n)=prime(17*n) \\ Edward Jiang, Sep 08 2014

[nth_prime(17*n) for n in (1..50)] # Bruno Berselli, May 07 2014

cp's OEIS Frontend

A140444 Primes congruent to 1 (mod 14).

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A140542 Primes of form 17*n - 6.

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A140545 Primes of form 17n + 6.

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A092074 Primes congruent to 3 mod 17.

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A138631 Primes of the form 17*k + 9.

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A141865 Primes congruent to 13 mod 17.

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A138633 Primes of the form 17*k - 9.

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