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A002496 Primes of form n^2 + 1.
(Formerly M1506 N0592)
127
2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

An equivalent description: primes of form P = (p1*p2*...*pm)^k + 1 where p1 ... pm are primes and k>1, since then k must be even for P 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 Elemer, 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, Jun 05 2004

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). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003

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, Mar 04 2007

With exception of two first terms {2,5} continued fraction (1+sqrt(p))/2 has period 3. - Artur Jasinski, Feb 03 2010

With exception of the first term {2} congruent to 1 mod 4.

With exception of the first two terms, congruent to 1 or 17 mod 20. - Robert Israel, Oct 14 2014

REFERENCES

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.

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

LINKS

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.

P. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, 16 (1962), 363-367.

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

Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, Amer. Math. Monthly 56 (1949), 17-19.

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

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

Wikipedia, Bateman-Horn Conjecture

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

FORMULA

There are O(sqrt(n)/log(n)) members of this sequence up to n. But this is just an upper bound. See the Bateman-Horn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.

MAPLE

select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # Robert Israel, Oct 14 2014

MATHEMATICA

Select[Range[100]^2+1, PrimeQ]

PROG

(PARI) isA002496(n) = isprime(n) && issquare(n-1) \\ Michael B. Porter, Mar 21 2010

(PARI) is_A002496(n)=issquare(n-1)&&isprime(n) \\ For "random" numbers in the range 10^10 and beyond, at least 5 times faster than the above. - M. F. Hasler, Oct 14 2014

(MAGMA) [p: p in PrimesUpTo(100000)| IsSquare(p-1)]; // Vincenzo Librandi, Apr 09 2011

(Haskell)

a002496 n = a002496_list !! (n-1)

a002496_list = filter ((== 1) . a010051') a002522_list

-- Reinhard Zumkeller, May 06 2013

(Python)

# Python 3.2 or higher required

from itertools import accumulate

from sympy import isprime

A002496_list = [n+1 for n in accumulate(range(10**5), lambda x, y:x+2*y-1) if isprime(n+1)] # Chai Wah Wu, Sep 23 2014

CROSSREFS

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

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

Cf. A000668 = Mersenne primes.

Subsequence of A039770.

Cf. A010051, subsequence of A002522.

Cf. A237040 (an analog for n^3 + 1).

Sequence in context: A078523 A078324 A240322 * A127436 A064168 A118727

Adjacent sequences:  A002493 A002494 A002495 * A002497 A002498 A002499

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formula, reference, and comment from Charles R Greathouse IV, Aug 24 2009

Edited by M. F. Hasler, Oct 14 2014

STATUS

approved

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Last modified October 21 03:30 EDT 2014. Contains 248371 sequences.