A027861 Numbers k such that k^2 + (k+1)^2 is prime.

1, 2, 4, 5, 7, 9, 12, 14, 17, 19, 22, 24, 25, 29, 30, 32, 34, 35, 39, 42, 47, 50, 60, 65, 69, 70, 72, 79, 82, 84, 85, 87, 90, 97, 99, 100, 102, 104, 109, 110, 115, 122, 130, 135, 137, 139, 144, 149, 154, 157, 160, 162, 164, 167, 172, 174, 185, 187, 189, 195, 199, 202

Offset

1,2

Comments

k > 1 never ends in 1, 3, 6 or 8 (that is, k*(k+1) does not end in 2). - Lekraj Beedassy, Jul 09 2004

k > 1 can never be congruent to (1 or 3) mod 5, because if it were, then k^2 + (k+1)^2 would be divisible by 5. In other words, for k > 1, this sequence cannot contain any values in A047219. This means that we can immediately discard 40% of all possible k. - Dmitry Kamenetsky, Sep 02 2008

The number of primes of this type is roughly twice the number of primes of the form k^2+1 (A005574), and bigger than the number of primes (A000040) up to same k. For example: pi_{k^2+1}(10^8) = 3954181 < pi(10^8) = pi_{k}(10^8) = 5761455 < pi_{k^2+(k+1)^2}(10^8) = 7755330. - Hermann Stamm-Wilbrandt, Dec 22 2025

Links

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

Formula

a(n) = (A002731(n)-1)/2.

a(n) = (sqrt(2*A027862(n)-1)-1)/2. - Zak Seidov, Jul 22 2013

A010051(A001844(a(n))) = 1. - Reinhard Zumkeller, Jul 13 2014

a(n) = floor(sqrt(A027862(n)/2)). - Rémi Guillaume, Apr 02 2025

Mathematica

Select[Range[250], PrimeQ[#^2+(#+1)^2]&] (* Harvey P. Dale, Dec 31 2017 *)

Prog

(Magma) [n: n in [0..1000] |IsPrime(n^2 + (n+1)^2)]; // Vincenzo Librandi, Nov 19 2010
(Haskell)
a027861 n = a027861_list !! (n-1)
a027861_list = filter ((== 1) . a010051 . a001844) [0..]
-- Reinhard Zumkeller, Jul 13 2014
(PARI) is(n)=isprime(n^2 + (n+1)^2) \\ Charles R Greathouse IV, Apr 28 2015

Crossrefs

Complement of A012132.

Cf. A002731 (2k+1 values), A027862 (resulting primes), A091277 (indices of resulting primes).

Cf. A047219 (k mod 5 = 1 or 3), A001844 (centered squares), A010051.

Cf. A005574 (k^2+1 prime), A000040 (primes).

Sequence in context: A286753 A325543 A214051 * A219648 A062428 A250000

Adjacent sequences: A027858 A027859 A027860 * A027862 A027863 A027864

Keyword

nonn,easy

Author

Patrick De Geest

Status

approved