A005574 Numbers k such that k^2 + 1 is prime. (Formerly M1010)

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396

Offset

1,2

Comments

Hardy and Littlewood conjectured that the asymptotic number of elements in this sequence not exceeding n is approximately c*sqrt(n)/log(n) for some constant c. - Stefan Steinerberger, Apr 06 2006

Also, nonnegative integers such that a(n)+i is a Gaussian prime. - Maciej Ireneusz Wilczynski, May 30 2011

Apparently Goldbach conjectured that any a>1 from this sequence can be written as a=b+c where b and c are in this sequence (Lemmermeyer link below). - Jeppe Stig Nielsen, Oct 14 2015

No term > 2 can be both in this sequence and in A001105 because of the Aurifeuillean factorization (2*k^2)^2 + 1 = (2*k^2 - 2*k + 1) * (2*k^2 + 2*k + 1). - Jeppe Stig Nielsen, Aug 04 2019

References

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.

R. K. Guy, "Unsolved Problems in Number Theory", 3rd edition, A2

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 15, Thm. 17.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

H. Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)

F. Ellermann, Primes of the form (m^2)+1 up to 10^6.

L. Euler, Lettre CXLIX (to Goldbach), 1752.

L. Euler, De numeris primis valde magnis, Novi Commentarii academiae scientiarum Petropolitanae 9 (1764), pp. 99-153. See pp. 123-125.

R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).

F. Lemmermeyer, Primes of the form a^2+1, Math Overflow question (2010).

Eric Weisstein's World of Mathematics, Landau's Problems.

Eric Weisstein's World of Mathematics, Power.

Eric Weisstein's World of Mathematics, Near-Square Prime.

Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Formula

a(n) = A090693(n) - 1.

a(n) = 2*A001912(n-1) for n>1. - Jeppe Stig Nielsen, Aug 04 2019

Mathematica

Select[Range[350], PrimeQ[ #^2 + 1] &] (* Stefan Steinerberger, Apr 06 2006 *)
Join[{1}, 2Flatten[Position[PrimeQ[Table[x^2+1, {x, 2, 1000, 2}]], True]]]  (* Fred Patrick Doty, Aug 18 2017 *)

Prog

(PARI) isA005574(n) = isprime(n^2+1) \\ Michael B. Porter, Mar 20 2010
(PARI) for(n=1, 1e3, if(isprime(n^2 + 1), print1(n, ", "))) \\ Altug Alkan, Oct 14 2015
(Magma) [n: n in [0..400] | IsPrime(n^2+1)]; // Vincenzo Librandi, Nov 18 2010
(Haskell)
a005574 n = a005574_list !! (n-1)
a005574_list = filter ((== 1) . a010051' . (+ 1) . (^ 2)) [0..]
-- Reinhard Zumkeller, Jul 03 2015

Crossrefs

Cf. A002522, A001912, A002496, A062325, A090693, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Other sequences of the type "Numbers k such that k^2 + i is prime": this sequence (i=1), A067201 (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

Cf. A010051, A259645, A295405 (characteristic function).

Sequence in context: A104692 A066755 A089238 * A109807 A259645 A191113

Adjacent sequences: A005571 A005572 A005573 * A005575 A005576 A005577

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved