A202018 a(n) = n^2 + n + 41.

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601, 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393

Offset

0,1

Comments

Euler's famous prime-generating polynomial; a(0) through a(39) are all prime.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = A005846(n) for n < 41, a(41) = A145292(1);

Union of A005846 (primes) and A145292 (composites);

a(n) = A002378(n) + 41.

a(a(n) + n) = a(n)*a(n+1). - Vladimir Shevelev, Jul 16 2012 (This identity holds for all sequences of the form n^2 + n + c, Joerg Arndt, Jul 17 2012).

a(0) = 41 and for n > 0, a(n) = a(n-1) + 2*n. - Jean-Christophe Hervé, Sep 27 2014

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Colin Barker, Sep 28 2014

G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. - Colin Barker, Sep 28 2014

a(n) = 2*a(n-1) - a(n-2) + 2. - Vincenzo Librandi, Mar 04 2016

E.g.f.: (x^2 + 2*x + 41)*exp(x). - Robert Israel, Mar 10 2016

Maple

A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # Wesley Ivan Hurt, Sep 28 2014

Mathematica

Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011

Prog

(Haskell)
a202018 = (+ 41) . a002378
(PARI) a(n)=n^2+n+41 \\ Charles R Greathouse IV, Dec 08 2011
(Magma) [n^2 + n + 41 : n in [0..50]]; // Wesley Ivan Hurt, Sep 28 2014
(Scala) (0 to 49).map((n: Int) => n * n + n + 41) // Alonso del Arte, Nov 29 2018
(Sage) [n^2+n+41 for n in range(50)] # G. C. Greubel, Dec 04 2018
(GAP) List([0..50], n -> n^2 +n+41); # G. C. Greubel, Dec 04 2018

Crossrefs

Cf. A060566, A010051, A000040, A002808.

Cf. A002378, A005846, A145292.

Cf. A056561.

Sequence in context: A257362 A330673 A296921 * A005846 A273756 A154498

Adjacent sequences: A202015 A202016 A202017 * A202019 A202020 A202021

Keyword

nonn,easy

Author

Reinhard Zumkeller, Dec 08 2011

Status

approved