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A216270 Numbers n such that n+(n+1), n^2+(n+1)^2, n+(n+1)^2, n^2+(n+1) are all prime. 1
1, 2, 5, 14, 69, 99, 495, 1364, 1365, 2010, 2735, 3099, 3914, 4359, 4389, 5984, 6669, 8435, 9164, 10794, 12075, 15224, 15315, 16014, 16470, 17900, 20214, 20769, 21204, 23450, 24240, 26430, 26690, 27300, 29099, 35189, 38415, 38745, 42944, 47054, 48789, 50295 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

Joong Fang, Abstract Algebra, Schaum, 1963, Page 76.

LINKS

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

EXAMPLE

n=14:                               29│     │421

n+(n+1)=14+(14+1)=29                   14---196

n^2+(n+1)^2=196+225=421                │  X  │

n+(n+1)^2=14+225=239                   15---225        *15+225+1=241

n^2+(n+1)=196+15=211               211/        \239

.

n=5:                                  11│   │61

n+(n+1)=5+(5+1)=11                      5---25

n^2+(n+1)^2=25+36=61                    │ X │

n+(n+1)^2=5+36=41                       6---36          *6+36+1=43

n^2+(n+1)=25+6=31                    31/      \41

.

n=495:                             991│     │491041

n+(n+1)=495+496=991                   495---245025

n^2+(n+1)^2=491041                    │  X  │

n+(n+1)^2=246511                      496---246016

n^2+(n+1)=245521               245521/       \246511

.

They form the group:

o 1 2 3 (i)

1 0 3 2

2 3 1 0

3 2 0 1

.

For example, for n=99:

99   9801       0 1 2 3 (i)

100  10000

9801  99        1 0 3 2

10000 100

10000 100

99    9801      2 3 1 0

100  10000      3 2 0 1

9801 99

The sum of each column and the sum of each diagonal is a prime number.

MATHEMATICA

Select[Range[51000], AllTrue[{#+(#+1), #^2+(#+1)^2, #+(#+1)^2, #^2+#+1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 21 2017 *)

PROG

(PARI)

is(n) = { isprime(n+(n+1)) & isprime(n^2+(n+1)^2) & isprime(n+(n+1)^2) & isprime(n^2+(n+1)); }

for(n=1, 10^6, if (is(n), print1(n, ", ")));

/* Joerg Arndt, Mar 26 2013 */

CROSSREFS

Sequence in context: A227365 A059958 A102019 * A214374 A284661 A097595

Adjacent sequences:  A216267 A216268 A216269 * A216271 A216272 A216273

KEYWORD

nonn

AUTHOR

César Aguilera, Mar 15 2013

STATUS

approved

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Last modified June 25 08:27 EDT 2021. Contains 345453 sequences. (Running on oeis4.)