A124485 -id:A124485 - cp's OEIS frontend

A124488 Numbers k such that 2k-1, 4k-1, 6k-1, 8k-1 and 10*k-1 are primes.

3, 45, 6450, 18000, 22785, 41790, 54180, 77385, 87675, 98385, 103005, 104520, 151515, 187005, 210210, 244590, 256620, 320775, 329175, 354795, 382875, 387975, 431385, 495540, 509355, 528510, 632775, 763815, 804870, 810540, 812175, 849285

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A124485-A124492.

Select[Range[900000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10}*# - 1) &] (* Ray Chandler, Nov 21 2006 *)

A124489 Numbers k such that 2k-1, 4k-1, 6k-1, 8k-1, 10k-1 and 12k-1 are primes.

Original entry on oeis.org

77385, 87675, 320775, 329175, 509355, 1137045, 1447110, 2623005, 3310965, 3974880, 4095000, 4339335, 5183220, 6163815, 6975780, 9080190, 9462285, 10957170, 11139975, 11148900, 12382755, 12796140, 15514695, 15917580

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A124485-A124492.

Select[3*5*7*Range[160000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12}*# - 1) &] (* Ray Chandler, Nov 21 2006 *)
Select[Range[16*10^6],AllTrue[2*Range[6]#-1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 19 2019 *)

A124491 Numbers n such that 2n-1, 4n-1, 6n-1, 8n-1, 10n-1, 12n-1, 14n-1 and 16n-1 are primes.

Original entry on oeis.org

1447110, 30020760, 32261985, 121012185, 203937090, 546020475, 546037695, 837344865, 1140530160, 2517567255, 2664703335, 2841691440, 2896212165, 3000108405, 3190108740, 5204790360, 5744351970, 6872932605, 7090912185

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

All terms are multiples of 105. - Harvey P. Dale, May 08 2019

Cf. A124485-A124492.

k = 0;inc = 3*5*7;While[k < 7200000000,k += inc;While[Nand @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12, 14, 16}*k - 1), k += inc];Print[k];];(* Ray Chandler, Nov 21 2006 *)
Select[Range[105,7091*10^6,105],AllTrue[2*Range[8]#-1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 08 2019 *)

Corrected and extended by Ray Chandler, Nov 21 2006

A176045 Numbers n such that n-1 and 2*n-1 are both prime.

Original entry on oeis.org

3, 4, 6, 12, 24, 30, 42, 54, 84, 90, 114, 132, 174, 180, 192, 234, 240, 252, 282, 294, 360, 420, 432, 444, 492, 510, 594, 642, 654, 660, 684, 720, 744, 762, 810, 912, 954, 1014, 1020, 1032, 1050, 1104, 1224, 1230, 1290, 1410, 1440, 1452, 1482, 1500, 1512, 1560

Offset: 1

Michel Lagneau, Apr 07 2010

nonn

Also numbers n such that all eigenvalues of the n X n matrix M_n defined in A176043 are prime. The eigenvalues are 2*n-1, and n-1 with multiplicity n-1.

a(n)^2 = p^2 + q, where both p and q are primes. These are the only squares of this form, and which always yields q > p with a(n) - 1 = p = A005384(n) and 2*a(n) - 1 = q = A005385(n), for the same n. Also: a(n) = q - p; p + q + a(n) = 2q = A194593(n+1); and p*q = A156592 - Richard R. Forberg, Mar 04 2015

			6-1 = 5 and 2*6-1 = 11 are both prime, so 6 is in the sequence. 7-1 = 6 and 2*7-1 = 13 are not both prime, so 7 is not in the sequence.
p = 3, q = 7; p^2 + q = 16, a(n) = sqrt(16) = 4. - _Richard R. Forberg_, Mar 04 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A176043, A005384 (Sophie Germain primes), A005385 (Safe Primes), A124485 (2*n-1 and 4*n-1 are prime).

[ n: n in [2..1600] | IsPrime(n-1) and IsPrime(2*n-1) ]; // Klaus Brockhaus, Apr 19 2010

with(numtheory):for n from 2 to 2000 do:if type((2*n-1),prime)=true and type((n-1),prime)=true then print(n):else fi:od:

Select[Prime[Range[250]],PrimeQ[2#+1]&]+1 (* Harvey P. Dale, Jul 31 2013 *)

isok(n) = isprime(n-1) && isprime(2*n-1); \\ Michel Marcus, Apr 06 2016

a(n) = A005384(n)+1.

a(n) = 2*A124485(n-1) for n > 1.

Edited and 1482 inserted by Klaus Brockhaus, Apr 19 2010

A124490 Numbers n such that 2n-1, 4n-1, 6n-1, 8n-1, 10n-1, 12n-1 and 14n-1 are primes.

Original entry on oeis.org

1447110, 2623005, 4095000, 4339335, 6975780, 9080190, 12382755, 19455975, 20322960, 30020760, 32261985, 54202995, 62014155, 63196350, 66383520, 71369340, 94571295, 121012185, 124225920, 162780660, 177109380, 196068180, 223888665, 303047745, 310143960

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Cf. A124485-A124492.

Select[3*5*7*Range[2000000], And @@ PrimeQ /@ ({2, 4, 6, 8, 10, 12, 14}*# - 1) &] (* Ray Chandler, Nov 21 2006 *)
pr7Q[n_]:=AllTrue[2*Range[7]n-1,PrimeQ]; Select[105*Range[3*10^6],pr7Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 17 2018 *)

Extended by Ray Chandler, Nov 21 2006

More terms from Harvey P. Dale, May 17 2018

A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table.

Original entry on oeis.org

2, 4, 8, 10, 14, 18, 28, 30, 38, 44, 58, 60, 64, 78, 80, 84, 94, 98, 120, 140, 144, 148, 164, 170, 198, 214, 218, 220, 228, 240, 248, 254, 270, 304, 318, 338, 340, 344, 350, 368, 408, 410, 430, 470, 480, 484, 494, 500, 504, 520, 528, 534, 578, 604, 630, 634, 644, 658

Offset: 1

Hilko Koning, Aug 02 2018

nonn

In the first quadrant of a coordinate system define a rectangular Sophie Germain billiard table with width p and length 2p+1, with vertices (0,0), (p,0), (p,2p+1) and (0,2p+1). A billiard ball (considered to be a point) starts from (0,0) at an angle of 45 degrees and hits the sides exactly p times until it hits the x-axis. The sequence gives the intersections with the x-axis of consecutive Sophie Germain prime numbers (p > 3) after p bounces.
The sum of all crossed lattice points (including the rectangle sides) is the sum of crossed points left under, right middle and left up respectively ((p+7)/6)^2 + (p+1)(p+4)/18 + (p+1)(p+7)/36 = ((p+4)/3)^2 (see bouncing examples).
The enclosed areas in the Sophie Germain billiard table also correspond to ((p+4)/3)^2.
The number of trajectories is a subsequence of A176045.
The number of trajectories with slope +1 or with slope -1 is a subsequence of A124485.
The sum of a term of this sequence and the corresponding Sophie Germain prime is A317510 and it appears that this is a subsequence of A179882. Checked up to and including 33295 of A317510 (Sophie Germain prime 24971).

Samuel King, Billiard Simulator
Hilko Koning, Some bouncing examples.

Cf. A005384, A151922, A176045, A124485, A179882, A317510.

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (p + 1)/3]], {p, 5, 2*10^3}]; lst
(Select[Prime@ Range[3, 300], PrimeQ[2# + 1] &] + 1)/3 (* Robert G. Wilson v, Aug 02 2018 *)

lista(nn) = forprime(p=2, nn, if (isprime(2*p+1), print1((p+1)/3, ", "));); \\ Michel Marcus, Aug 25 2018

a(n) = (A005384(n)+1)/3 for n>=3. - Michel Marcus, Aug 25 2018

cp's OEIS Frontend

A124488 Numbers k such that 2*k-1, 4*k-1, 6*k-1, 8*k-1 and 10*k-1 are primes.

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A124489 Numbers k such that 2*k-1, 4*k-1, 6*k-1, 8*k-1, 10*k-1 and 12*k-1 are primes.

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A124491 Numbers n such that 2n-1, 4n-1, 6n-1, 8n-1, 10n-1, 12n-1, 14n-1 and 16n-1 are primes.

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A176045 Numbers n such that n-1 and 2*n-1 are both prime.

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A124490 Numbers n such that 2n-1, 4n-1, 6n-1, 8n-1, 10n-1, 12n-1 and 14n-1 are primes.

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A317626 Intersections with the x-axis of a bouncing ball on a Sophie Germain billiard table.

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A124488 Numbers k such that 2k-1, 4k-1, 6k-1, 8k-1 and 10*k-1 are primes.

A124489 Numbers k such that 2k-1, 4k-1, 6k-1, 8k-1, 10k-1 and 12k-1 are primes.