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
References
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 225.
- Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 138-139, 145.
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).
Programs
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GAP
List([0..50], n -> n^2 +n+41); # G. C. Greubel, Dec 04 2018
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Haskell
a202018 = (+ 41) . a002378
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Magma
[n^2 + n + 41 : n in [0..50]]; // Wesley Ivan Hurt, Sep 28 2014
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Maple
A202018:=n->n^2+n+41: seq(A202018(n), n=0..50); # Wesley Ivan Hurt, Sep 28 2014
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Mathematica
Table[n^2 + n + 41, {n, 0, 49}] (* Alonso del Arte, Dec 08 2011 *) -
PARI
a(n)=n^2+n+41 \\ Charles R Greathouse IV, Dec 08 2011
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Sage
[n^2+n+41 for n in range(50)] # G. C. Greubel, Dec 04 2018
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Scala
(0 to 49).map((n: Int) => n * n + n + 41) // Alonso del Arte, Nov 29 2018
Formula
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
From Colin Barker, Sep 28 2014: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (41*x^2 - 80*x + 41) / (1-x)^3. (End)
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
Sum_{n>=0} 1/a(n) = tanh(sqrt(163)*Pi/2)*Pi/sqrt(163). - Amiram Eldar, May 12 2025
Comments