A056561 Numbers n such that n^2 + n + 41 (known as Euler's polynomial) is prime.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78

Offset

1,3

Comments

Among first 100000 terms, the only run of 13 subsequent values >39 is 219..231. - Zak Seidov, Jan 28 2009

Number of terms less than 10^n: 1, 10, 86, 581, 4149, 31985, 261081, 2208197, 19132652, ... . - Robert G. Wilson v, Apr 20 2015

Complement of A007634. - Robert Israel, Apr 20 2015

References

P. Hoffman, Archimedes' Revenge, pp. 39-40,Penguin Books 1988.

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Euler Prime

Formula

a(n) = (sqrt(4*A005846(n)-163)-1)/2.

a(n) = A002837(n+1)-1. - Robert Price, Nov 08 2019

Example

39 is in the sequence because 39^2+39+41=1601 which is prime but 40 is not because 40^2+40+41=1681=41*41.

Maple

select(t -> isprime(t^2+t+41), [$0..100]); # Robert Israel, Apr 20 2015

Mathematica

Select[Range[80], PrimeQ[#^2 + # + 41] &] (* Vincenzo Librandi, Sep 28 2012 *)

Prog

(Magma) [n: n in [0..80] |IsPrime(n^2 + n + 41)]; // Vincenzo Librandi, Sep 28 2012
(Haskell)
a056561 n = a056561_list !! (n-1)
a056561_list = filter ((== 1) . a010051' . a202018) [0..]
-- Reinhard Zumkeller, Jul 03 2015
(PARI) is(n)=isprime(n^2+n+41) \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

Cf. A002837, A005846, A007634, A010051, A202018, A259645.

Sequence in context: A366188 A002837 A271143 * A321993 A236674 A273886

Adjacent sequences: A056558 A056559 A056560 * A056562 A056563 A056564

Keyword

nonn,easy

Author

Henry Bottomley, Jun 26 2000

Extensions

Definition amended by Harvey P. Dale, Sep 13 2025

Status

approved