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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
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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Simpler definition from N. J. A. Sloane, Jul 11 2008
STATUS
approved

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Last modified March 5 03:04 EST 2024. Contains 370537 sequences. (Running on oeis4.)