A090614 -id:A090614 - cp's OEIS frontend

A045437 Primes congruent to 3 mod 7.

3, 17, 31, 59, 73, 101, 157, 199, 227, 241, 269, 283, 311, 353, 367, 409, 479, 521, 563, 577, 619, 647, 661, 773, 787, 829, 857, 941, 983, 997, 1039, 1109, 1123, 1151, 1193, 1249, 1277, 1291, 1319, 1361, 1459, 1487, 1543, 1571, 1613, 1627, 1669, 1697, 1753

Offset: 1

N. J. A. Sloane

Also primes congruent to 3 mod 14.

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

Primes arising in sequences A024903, A033868, A089033, A090614.

A090613 gives prime index.

[p: p in PrimesUpTo(2000) | p mod 7 eq 3]; // Vincenzo Librandi, Aug 06 2012

Select[Range[3, 50000, 7], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)

is(n)=isprime(n) && n%7==3 \\ Charles R Greathouse IV, Jul 01 2016

Extended by Ray Chandler, Dec 23 2003

A089033 Numbers n such that 7*n+3 is prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 22, 28, 32, 34, 38, 40, 44, 50, 52, 58, 68, 74, 80, 82, 88, 92, 94, 110, 112, 118, 122, 134, 140, 142, 148, 158, 160, 164, 170, 178, 182, 184, 188, 194, 208, 212, 220, 224, 230, 232, 238, 242, 250, 260, 268, 272, 278, 298, 304, 320, 334, 340

Offset: 1

Giovanni Teofilatto, Nov 12 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

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

A045437 gives primes, A090613 gives prime index.

Cf. A024903, A033868, A090614.

Mathematica
```
Select[Range[1000],PrimeQ[7#+3]&]
```

is(n)=isprime(7*n+3) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A024903(n)-1 = A033868(n)-2 = A090614(n)*2.

Corrected and extended by Ray Chandler, Nov 12 2003

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A024903 Numbers k such that 7*k - 4 is prime.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 23, 29, 33, 35, 39, 41, 45, 51, 53, 59, 69, 75, 81, 83, 89, 93, 95, 111, 113, 119, 123, 135, 141, 143, 149, 159, 161, 165, 171, 179, 183, 185, 189, 195, 209, 213, 221, 225, 231, 233, 239, 243, 251, 261, 269, 273, 279, 299, 305, 321, 335, 341

Offset: 1

Clark Kimberling

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

Cf. A045437 (associated primes), A090613 (gives prime index).

Cf. A033868, A089033, A090614.

[n: n in [1..400] | IsPrime(7*n-4)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[350], PrimeQ[7 # - 4] &] (* Vincenzo Librandi, Sep 26 2012 *)

is(n)=isprime(7*n-4) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A033868(n-1)-1 = A089033(n-1)+1 = A090614(n-1)*2+1.

A090613 Numbers k such that the k-th prime is congruent to 3 mod 7.

Original entry on oeis.org

2, 7, 11, 17, 21, 26, 37, 46, 49, 53, 57, 61, 64, 71, 73, 80, 92, 98, 103, 106, 114, 118, 121, 137, 138, 145, 148, 160, 166, 168, 175, 186, 188, 190, 196, 204, 206, 210, 215, 218, 232, 236, 243, 248, 255, 258, 263, 265, 273, 281, 289, 292, 296, 316, 321, 334

Offset: 1

Ray Chandler, Dec 23 2003

nonn

Also numbers k such that the k-th prime is congruent to 3 mod 14.
A045437 indexed by A000040.
The asymptotic density of this sequence is 1/6 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Cf. A024903, A033868, A045437, A089033, A090614.

Select[Range[350], Mod[Prime[#], 7] == 3 &] (* Amiram Eldar, Mar 01 2021 *)

a(n) = k such that A000040(k) = A045437(n).

Offset corrected by Amiram Eldar, Mar 01 2021

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

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

A033868 Numbers n such that 7*n-11 is prime.

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 24, 30, 34, 36, 40, 42, 46, 52, 54, 60, 70, 76, 82, 84, 90, 94, 96, 112, 114, 120, 124, 136, 142, 144, 150, 160, 162, 166, 172, 180, 184, 186, 190, 196, 210, 214, 222, 226, 232, 234, 240, 244, 252, 262, 270, 274, 280, 300, 306, 322, 336, 342

Offset: 1

Giovanni Teofilatto, Dec 01 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

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

Cf. A045437 (associated primes), A090613 (gives prime index).

Cf. A024903, A089033, A090614.

[n: n in [0..350] | IsPrime(7*n - 11)]; // Vincenzo Librandi, Sep 26 2012

Select[Range[500],PrimeQ[7#-11]&]  (* Harvey P. Dale, Mar 31 2011 *)

is(n)=isprime(7*n-11) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = A024903(n)+1 = A089033(n)+2 = A090614(n)*2+2.

Extended by Ray Chandler, Dec 23 2003

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A140442 Primes congruent to 9 mod 14.

Original entry on oeis.org

23, 37, 79, 107, 149, 163, 191, 233, 317, 331, 359, 373, 401, 443, 457, 499, 541, 569, 653, 709, 751, 821, 863, 877, 919, 947, 1031, 1087, 1129, 1171, 1213, 1283, 1297, 1367, 1381, 1409, 1423, 1451, 1493, 1549, 1619, 1759, 1787, 1801, 1871, 1913, 1997

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

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

Primes arising in sequences A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.

[p: p in PrimesUpTo(2000)|p mod 14 in {9}] // Vincenzo Librandi, Dec 18 2010

Select[Prime[Range[500]],MemberQ[{9},Mod[#,14]]&] (* Vincenzo Librandi, Aug 07 2012 *)
Select[Range[9,2000,14],PrimeQ] (* Harvey P. Dale, Apr 17 2015 *)

is(n)=isprime(n) && n%14==9 \\ Charles R Greathouse IV, Jul 03 2016

a(n) = A045392(n+1) = A045383(n+2). - Zak Seidov, Mar 12 2014

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 03 2016

1451 inserted by R. J. Mathar, Sep 13 2008

A153307 Numbers n such that 14*n+3 is not prime.

Original entry on oeis.org

3, 6, 8, 9, 10, 12, 13, 15, 18, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 42, 43, 45, 48, 49, 50, 51, 52, 53, 54, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 72, 73, 75, 76, 77, 78, 81, 83, 84, 86, 87, 88, 90

Offset: 1

Vincenzo Librandi, Dec 23 2008

			Distribution of the terms in the following triangular array:
*;
*,*;
*,*,*;
*,3,*,*;
*,*,*,*,*;
*,*,*,*,10,*;
3,*,*,*,*,*,*;
*,*,*,*,*,*,18,*;
*,*,*,12,*,*,*,*,*;
*,*,*,*,*,*,*,*,*,*;
*,8,*,*,*,*,*,*,31,*,*;
*,*,*,*,*,23,*,*,*,*,*;
*,*,*,*,21,*,*,*,*,*,*,48,*;etc.
where * marks the non-integer values of (2*h*k + k + h - 1)/7 with h >= k >= 1. - _Vincenzo Librandi_, Jan 17 2013

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

A090614.

[n: n in [0..150] | not IsPrime(14*n + 3)]; // Vincenzo Librandi, Jan 12 2013

Select[Range[0, 200], !PrimeQ[14*# + 3]&] (* Vincenzo Librandi, Jan 12 2013 *)

cp's OEIS Frontend

A045437 Primes congruent to 3 mod 7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A089033 Numbers n such that 7*n+3 is prime.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A024903 Numbers k such that 7*k - 4 is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A090613 Numbers k such that the k-th prime is congruent to 3 mod 7.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A140444 Primes congruent to 1 (mod 14).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

A033868 Numbers n such that 7*n-11 is prime.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A140442 Primes congruent to 9 mod 14.

Views

Author

Keywords

Links

Crossrefs