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 A027862 Primes of the form n^2 + (n+1)^2. 40
 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 (m^2+1)/2. - Zak Seidov, May 01 2014 Note that ((2n+1)^2 + 1)/2 = n^2 + (n+1)^2. - Thomas Ordowski, May 25 2015 Primes p such that 2p-1 is a square. - Thomas Ordowski, Aug 27 2016 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. Daniel Shanks, An analytic criterion for the existence of infinitely many primes of the form 1/2 * (n^2 + 1), Illinois Journal of Mathematics 8:3 (1964), p. 377-379. LINKS T. D. Noe and Zak Seidov, Table of n, a(n) for n = 1..10000 P. De Geest, World!Of Numbers W. SierpiĆski, 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 EXAMPLE 13 is in the sequence because it is prime and 13 = 2^2 + 3^2. - Michael B. Porter, Aug 27 2016 MATHEMATICA Select[Table[n^2+(n+1)^2, {n, 200}], PrimeQ] (* Harvey P. Dale, Aug 22 2012 *) Select[Total/@Partition[Range[200]^2, 2, 1], PrimeQ] (* Harvey P. Dale, Apr 20 2016 *) 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 * A308442 A322155 A100210 Adjacent sequences:  A027859 A027860 A027861 * A027863 A027864 A027865 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Cino Hilliard, Mar 16 2003 STATUS approved

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Last modified September 19 02:22 EDT 2020. Contains 337175 sequences. (Running on oeis4.)