1,1

It is conjectured that this sequence is infinite, but this has never been proved.

An equivalent description: primes of form m = (p1*p2*...*pm)^k + 1 where p1 ... pm are primes and k>1, since then k must be even for m to be prime.

Also prime = p(n) if A054269(n) = 1, i.e. quotient-cycle-length = 1 in continued fraction expansion of sqrt(p) - Labos E. (labos(AT)ana.sote.hu), Feb 21 2001

Also primes p such that phi(p) is a square.

Also primes of form x*y + z, where x, y and z are three successive numbers. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Jun 05 2004

From Pieter Moree (moree(AT)science.uva.nl), Nov 03 2003: It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A denotes the Artin constant (A = prod_q (1-1/(q(q-1))), q running over all primes). Numerically A = 0.3739558136.... More precisely, Sum_{p <= x} mu(p-1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: sum_{p <= x, mu(p-1)=1} 1 = (A/2)x/log x + o(x/log x) and sum_{p <= x, mu(p-1)=-1} 1 = (A/2)x/log x + o(x/log x).

Also primes of the form x^y + 1, where x>0, y>1. Primes of the form x^y - 1 (x>0, y>1) are the Mersenne primes listed in A000668(n) = {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Mar 04 2007

Also, if p=n^2+1, the congruence p^m=1 mod (n) is always verified for all m. Example: p=5, n=2, then 5^m=1 mod(2), 5=1 mod(2), 5^2=1 mod(2), 5^3=1 mod(2); p=37, n=6, then 37^3=1 mod (6); p=101, n=10, then 101^5=1 mod (10) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 05 2009]

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 211 pp. 34; 169, Ellipses Paris 2004.

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 22.

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

Leon Mirsky, Amer. Math. Monthly 56 (1949), 17-19.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.

C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134

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

W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.

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

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

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

Marek Wolf, Search for primes of the form m^2+1

There are O(sqrt(n)/log(n)) members of this sequence up to n.

Intersection[Table[n^2 + 1, {n, 225}], Prime[Range[5153]]]

s=""; For[i=1, i<10^2, If[PrimeQ[i^2+1], s=s<>ToString[i^2+1]<>", "]; i++ ]; Print[s] (from Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 29 2008)

Cf. A083844 (number of these primes < 10^n).

Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293.

Cf. A000668 = Mersenne primes.

Subsequence of A039770.

Sequence in context: A107630 A078523 A078324 this_sequence A127436 A064168 A118727

Adjacent sequences: A002493 A002494 A002495 this_sequence A002497 A002498 A002499

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Formula, reference, and comment from Charles R Greathouse IV (charles.greathouse(AT)case.edu), Aug 24 2009