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A140444 Primes congruent to 1 (mod 14). 7

%I #75 Sep 08 2022 08:45:34

%S 29,43,71,113,127,197,211,239,281,337,379,421,449,463,491,547,617,631,

%T 659,673,701,743,757,827,883,911,953,967,1009,1051,1093,1163,1289,

%U 1303,1373,1429,1471,1499,1583,1597,1667,1709,1723,1877,1933,2003,2017,2087

%N Primes congruent to 1 (mod 14).

%C From _Federico Provvedi_, May 24 2018: (start)

%C Also primes congruent to 1 (mod 7).

%C For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).

%C 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 P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x) (mod 29)

%C P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x) (mod 43)

%C P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)

%C P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x) (mod 113)

%C ...

%C (End).

%C Primes in A131877. - _Eric Chen_, Jun 14 2018

%H Vincenzo Librandi, <a href="/A140444/b140444.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) ~ 6n log n. - _Charles R Greathouse IV_, Jul 02 2016

%p select(isprime,select(n->modp(n,14)=1,[$1..2300])); # _Muniru A Asiru_, Jun 27 2018

%t Select[Prime[Range[500]], Mod[#, 14] == 1 &] (* _Harvey P. Dale_, Mar 21 2011 *)

%o (Magma) [p: p in PrimesUpTo(3000)|p mod 14 in {1}] // _Vincenzo Librandi_, Dec 18 2010]

%o (PARI) is(n)=isprime(n) && n%14==1 \\ _Charles R Greathouse IV_, Jul 02 2016

%o (GAP) Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # _Muniru A Asiru_, Jun 27 2018

%Y Cf. A045437, A045458, A045471, A045473.

%Y A090613 gives prime index.

%Y Cf. A090614.

%Y 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 nonn,easy

%O 1,1

%A _Juri-Stepan Gerasimov_, Jun 26 2008

%E Simpler definition from _N. J. A. Sloane_, Jul 11 2008

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)