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A140444 Primes congruent to 1 (mod 14). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

FORMULA

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

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 *)

PROG

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

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

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

CROSSREFS

Cf. A045437, A045458, A045471, A045473.

A090613 gives prime index.

Cf. A090614.

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).

Sequence in context: A316741 A173967 A004619 * A042969 A042967 A061638

Adjacent sequences:  A140441 A140442 A140443 * A140445 A140446 A140447

KEYWORD

nonn,easy

AUTHOR

Juri-Stepan Gerasimov, Jun 26 2008

EXTENSIONS

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

STATUS

approved

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Last modified September 20 22:20 EDT 2019. Contains 327252 sequences. (Running on oeis4.)