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A027862 Primes of the form n^2 + (n+1)^2. 30
5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, 4513, 5101, 7321, 8581, 9661, 9941, 10513, 12641, 13613, 14281, 14621, 15313, 16381, 19013, 19801, 20201, 21013, 21841, 23981, 24421, 26681 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also, primes of the form 4*k+1 which are the hypotenuse of one and only one right triangle with integral arms. - Cino Hilliard, Mar 16 2003

Centered square primes (i.e., prime terms of centered squares A001844). - Lekraj Beedassy, Jan 21 2005

Primes of the form 2*k*(k-1)+1. - Juri-Stepan Gerasimov, Apr 27 2010

Equivalently, primes of the form (m^2+1)/2 (take m=2*n+1). These primes a(n) have nontrivial solutions of x^2==1 (Modd a(n)) given by x=x(n)=A002731(n). For Modd n see a comment on A203571. See also A206549 for such solutions for primes of the form 4*k+1, given in A002144.

  E.g., a(3)=41, A002731(3)=9, 9^2=81, floor(81/41)=1 (odd),

  -81 = -2*41 + 1 == 1(mod 41), hence 9^2==1(Modd 41). - Wolfdieter Lang, Feb 24 2012

Also primes of the form 4*k+1 that are the smallest side length of one and only one integer Soddyian triangle (see A230812). - Frank M Jackson, Mar 13 2014

Also, primes of the form (n^2+1)/2. - Zak Seidov, May 01 2014

REFERENCES

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 271.

Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972. pp. 275.

LINKS

T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000

P. De Geest, World!Of Numbers

W. SierpiƄski, Sur les nombres triangulaires qui sont sommes de deux nombres triangulaires, Elem. Math., 17 (1962), pp. 63-65.

Panayiotis G. Tsangaris, A sieve for all primes of the form x^2 + (x+1)^2, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25 (1998), pp. 39-53.

FORMULA

a(n) = ((A002731(n)^2 - 1) / 2) + 1. - Torlach Rush, Mar 14 2014

a(n) = ((A002731(n)^2 + 1) / 2). - Zak Seidov, May 01 2014

MATHEMATICA

lst={}; Do[If[PrimeQ[p=n^2+(n+1)^2], AppendTo[lst, p]], {n, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

Select[Table[n^2+(n+1)^2, {n, 200}], PrimeQ] (* Harvey P. Dale, Aug 22 2012 *)

PROG

(PARI) je=[]; for(n=1, 500, if(isprime(n^2+(n+1)^2), je=concat(je, n^2+(n+1)^2))); je

(PARI) fermat(n) = { for(x=1, n, y=2*x*(x+1)+1; if(isprime(y), print1(y" ")) ) }

(MAGMA) [ a: n in [0..150] | IsPrime(a) where a is n^2+(n+1)^2 ]; // Vincenzo Librandi, Dec 18 2010

CROSSREFS

Primes p such that A079887(p) = 1.

Primes arising in A002731, A027861 gives n values, A091277 gives prime index.

Sequence in context: A087938 A103729 A234739 * A100210 A080267 A034735

Adjacent sequences:  A027859 A027860 A027861 * A027863 A027864 A027865

KEYWORD

nonn,easy,nice,changed

AUTHOR

Patrick De Geest

EXTENSIONS

More terms from Cino Hilliard, Mar 16 2003

STATUS

approved

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Last modified December 21 20:27 EST 2014. Contains 252326 sequences.