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

Author

Juri-Stepan Gerasimov, Jun 26 2008

Keywords

• nonn
• easy

Comments

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

Links

• 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.

Crossrefs

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).

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Formula

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

Extensions

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