This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A140444 #91 Oct 12 2024 02:53:03
%S A140444 29,43,71,113,127,197,211,239,281,337,379,421,449,463,491,547,617,631,
%T A140444 659,673,701,743,757,827,883,911,953,967,1009,1051,1093,1163,1289,
%U A140444 1303,1373,1429,1471,1499,1583,1597,1667,1709,1723,1877,1933,2003,2017,2087
%N A140444 Primes congruent to 1 (mod 14).
%C A140444 From _Federico Provvedi_, May 24 2018: (Start)
%C A140444 Also primes congruent to 1 (mod 7).
%C A140444 For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
%C A140444 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:
%C A140444 P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x) (mod 29)
%C A140444 P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x) (mod 43)
%C A140444 P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
%C A140444 P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x) (mod 113)
%C A140444 ... (End).
%C A140444 Primes in A131877. - _Eric Chen_, Jun 14 2018
%H A140444 Vincenzo Librandi, <a href="/A140444/b140444.txt">Table of n, a(n) for n = 1..1000</a>
%H A140444 Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, <a href="https://doi.org/10.7546/nntdm.2024.30.3.516-529">The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences</a>, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.
%F A140444 a(n) ~ 6n log n. - _Charles R Greathouse IV_, Jul 02 2016
%p A140444 select(isprime,select(n->modp(n,14)=1,[$1..2300])); # _Muniru A Asiru_, Jun 27 2018
%t A140444 Select[Prime[Range[500]], Mod[#, 14] == 1 &] (* _Harvey P. Dale_, Mar 21 2011 *)
%o A140444 (Magma) [p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // _Vincenzo Librandi_, Dec 18 2010
%o A140444 (PARI) is(n)=isprime(n) && n%14==1 \\ _Charles R Greathouse IV_, Jul 02 2016
%o A140444 (GAP) Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # _Muniru A Asiru_, Jun 27 2018
%Y A140444 Cf. A045437, A045458, A045471, A045473.
%Y A140444 A090613 gives prime index.
%Y A140444 Cf. A090614.
%Y A140444 Cf. A131877.
%Y A140444 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).
%K A140444 nonn,easy
%O A140444 1,1
%A A140444 _Juri-Stepan Gerasimov_, Jun 26 2008
%E A140444 Simpler definition from _N. J. A. Sloane_, Jul 11 2008