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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
OFFSET
1,2
REFERENCES
Joong Fang, Abstract Algebra, Schaum, 1963, Page 76.
LINKS
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
KEYWORD
nonn
AUTHOR
César Aguilera, Mar 15 2013
STATUS
approved