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A002496 Primes of the form k^2 + 1.
(Formerly M1506 N0592)
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)



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 the first two terms {2,5}, the 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). - Artur Jasinski, Mar 22 2011

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

From Bernard Schott, Mar 22 2019: (Start)

These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is square: A054755. So this sequence is a subsequence of A054755 and of A039770. Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.

If p prime = n^2 + 1, phi(p) = n^2 and cototient(p) = 1^2.

Except for 3, the four Fermat primes in A019434 {5, 17, 257, 65537}, belong to this sequence; with F_k = 2^(2^k) + 1, phi(F_k) = (2^(2^(k-1)))^2.

See the file "Subfamilies and subsequences" (& I) in A039770 for more details, proofs with data, comments, formulas and examples. (End)

In this sequence, primes ending with 7 seem to appear twice as often as primes ending with 1. This is because those with 7 come from integers ending with 4 or 6, while those with 1 come only from integers ending with 0 (see De Koninck & Mercier reference). - Bernard Schott, Nov 29 2020


Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 211 pp. 34 and 169, Ellipses, Paris, 2004.

Leonhard 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.

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

C. Stanley 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

Tewodros Amdeberhan, Luis A. Medina and Victor H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory, Vol. 128, No. 6 (2008), pp. 1807-1846, eq. (1.10).

William D. Banks, John B. Friedlander, Carl Pomerance and Igor 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.

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363-367.

N. A. Carella, The Euler Polynomial Prime Values Problem, arXiv:1912.05923 [math.GM], 2019.

Frank 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, Vol. 56, No. 1 (1949), pp. 17-19.

Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.

Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.

Apoloniusz Tyszka and Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).

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, arXiv:0803.1456 [math.NT], 2008-2010.


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.

a(n) = 1 + A005574(n)^2. - R. J. Mathar, Jul 31 2015

Sum_{n>=1} 1/a(n) = A172168. - Amiram Eldar, Nov 14 2020


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


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

Join[{2}, Select[Range[2, 300, 2]^2+1, PrimeQ]] (* Harvey P. Dale, Dec 18 2018 *)


(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


a002496 n = a002496_list !! (n-1)

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

-- Reinhard Zumkeller, May 06 2013


# 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


# Python 2.4 or higher required

from sympy import isprime

A002496_list = list(filter(isprime, (n*n+1 for n in range(10**5)))) # David Radcliffe, Jun 26 2016


Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).

Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293, A172168.

Cf. A000668 (Mersenne primes), A019434 (Fermat primes).

Subsequence of A039770.

Cf. A010051, subsequence of A002522.

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

Cf. A010051, A000290; subsequence of A028916.

Subsequence of A039770, A054754, A054755, A063752.

Primes of form n^2+b^4, b fixed: A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).

Cf. A030430 (primes ending with 1), A030432 (primes ending with 7).

Sequence in context: A078324 A240322 A276460 * A127436 A064168 A118727

Adjacent sequences:  A002493 A002494 A002495 * A002497 A002498 A002499




N. J. A. Sloane


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

Edited by M. F. Hasler, Oct 14 2014



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Last modified March 3 00:46 EST 2021. Contains 341756 sequences. (Running on oeis4.)