A113801 -id:A113801 - cp's OEIS frontend

A045472 Primes congruent to {1, 6} mod 7.

13, 29, 41, 43, 71, 83, 97, 113, 127, 139, 167, 181, 197, 211, 223, 239, 251, 281, 293, 307, 337, 349, 379, 419, 421, 433, 449, 461, 463, 491, 503, 547, 587, 601, 617, 631, 643, 659, 673, 701, 727, 743, 757, 769

Offset: 1

N. J. A. Sloane

Primes p such that p^4 = 1 mod 210. - Gary Detlefs, Dec 29 2011

Primes in A047336, also in A113801. - Reinhard Zumkeller, Jan 07 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A042989 (complement), A010051.

a045472 n = a045472_list !! (n-1)
a045472_list = [x | x <- a047336_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012

[ p: p in PrimesUpTo(1000) | p mod 7 in {1,6} ]; // Vincenzo Librandi, Aug 13 2012

Select[Prime[Range[200]],MemberQ[{1,6},Mod[#,7]]&] (* Vincenzo Librandi, Aug 13 2012 *)

select(p->abs(centerlift(Mod(p,7)))==1, primes(100)) \\ Charles R Greathouse IV, Mar 17 2022

a(n) ~ 3n log n. - Charles R Greathouse IV, Mar 17 2022

A219390 Numbers k such that 14*k+1 is a square.

Original entry on oeis.org

0, 12, 16, 52, 60, 120, 132, 216, 232, 340, 360, 492, 516, 672, 700, 880, 912, 1116, 1152, 1380, 1420, 1672, 1716, 1992, 2040, 2340, 2392, 2716, 2772, 3120, 3180, 3552, 3616, 4012, 4080, 4500, 4572, 5016, 5092, 5560, 5640, 6132, 6216, 6732, 6820

Offset: 1

Bruno Berselli, Nov 19 2012

Equivalently, numbers of the form m*(14*m+2), where m = 0,-1,1,-2,2,-3,3,...

Also, integer values of 2*h*(h+1)/7.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. similar sequences listed in A219257.

Cf. A113801 (square roots of 14*a(n)+1).

[n: n in [0..7000] | IsSquare(14*n+1)];

I:=[0,12,16,52,60]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A219390:=proc(q)
local n;
for n from 1 to q do if type(sqrt(14*n+1), integer) then print(n);
fi; od; end:
A219390(1000); # Paolo P. Lava, Feb 19 2013

Select[Range[0, 7000], IntegerQ[Sqrt[14 # + 1]] &]
CoefficientList[Series[4 x (3 + x + 3 x^2) ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,12,16,52,60},50] (* Harvey P. Dale, Feb 05 2019 *)

G.f.: 4*x^2*(3+x+3*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (14*n*(n-1)+5*(-1)^n*(2*n-1)+1)/4 +1.
a(n) = 2*A219191(n).
Sum_{n>=2} 1/a(n) = 7/2 - cot(Pi/7)*Pi/2. - Amiram Eldar, Mar 15 2022

A147832 Numbers congruent (0,2) mod 14.

Original entry on oeis.org

0, 2, 14, 16, 28, 30, 42, 44, 56, 58, 70, 72, 84, 86, 98, 100, 112, 114, 126, 128, 140, 142, 154, 156, 168, 170, 182, 184, 196, 198, 210, 212, 224, 226, 238, 240, 252, 254, 266, 268, 280, 282, 294, 296, 308, 310, 322, 324, 336, 338, 350, 352, 364, 366, 378, 380

Offset: 1

Giovanni Teofilatto, Nov 14 2008

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A113801, A047274, A001106.

Flatten[{#,#+2}&/@(14 Range[0,30])]  (* Harvey P. Dale, Dec 25 2010 *)

Except for the initial term, a(n) = A113801(n-1) + 1.
a(n) = 14*n - a(n-1) - 26 (with a(1)=0). - Vincenzo Librandi, Dec 17 2010
From Bruno Berselli, Dec 17 2010: (Start)
G.f.: 2*x^2*(1+6*x)/((1+x)*(1-x)^2).
a(n) = 2*A047274(n) = (14*n - 5*(-1)^n - 19)/2.
a(n) = 2*(A001106(n-1) - Sum_{i=1..n-1} a(i)) for n > 1. (End)

382 replaced with 380 by R. J. Mathar, Jun 28 2010

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A045472 Primes congruent to {1, 6} mod 7.

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A219390 Numbers k such that 14*k+1 is a square.

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A147832 Numbers congruent (0,2) mod 14.

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