%I M1506 N0592
%S 2,5,17,37,101,197,257,401,577,677,1297,1601,2917,3137,4357,5477,7057,
%T 8101,8837,12101,13457,14401,15377,15877,16901,17957,21317,22501,
%U 24337,25601,28901,30977,32401,33857,41617,42437,44101,50177
%N Primes of the form k^2 + 1.
%C It is conjectured that this sequence is infinite, but this has never been proved.
%C 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.
%C Also prime = p(n) if A054269(n) = 1, i.e., quotientcyclelength = 1 in continued fraction expansion of sqrt(p).  _Labos Elemer_, Feb 21 2001
%C Also primes p such that phi(p) is a square.
%C Also primes of form x*y + z, where x, y and z are three successive numbers.  _Giovanni Teofilatto_, Jun 05 2004
%C It is a result that goes back to Mirsky that the set of primes p for which p1 is squarefree has density A, where A denotes the Artin constant (A = prod_q (11/(q(q1))), q running over all primes). Numerically A = 0.3739558136.... More precisely, Sum_{p <= x} mu(p1)^2 = Ax/log x + o(x/log x) as x tends to infinity. Conjecture: Sum_{p <= x, mu(p1)=1} 1 = (A/2)x/log x + o(x/log x) and Sum_{p <= x, mu(p1)=1} 1 = (A/2)x/log x + o(x/log x).  Pieter Moree (moree(AT)mpimbonn.mpg.de), Nov 03 2003
%C 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
%C With exception of the first two terms {2,5}, the continued fraction (1 + sqrt(p))/2 has period 3.  _Artur Jasinski_, Feb 03 2010
%C With exception of the first term {2}, congruent to 1 (mod 4).  _Artur Jasinski_, Mar 22 2011
%C With exception of the first two terms, congruent to 1 or 17 (mod 20).  _Robert Israel_, Oct 14 2014
%C From _Bernard Schott_, Mar 22 2019: (Start)
%C 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.
%C If p prime = n^2 + 1, phi(p) = n^2 and cototient(p) = 1^2.
%C 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^(k1)))^2.
%C See the file "Subfamilies and subsequences" (& I) in A039770 for more details, proofs with data, comments, formulas and examples. (End)
%C 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
%D JeanMarie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 211 pp. 34 and 169, Ellipses, Paris, 2004.
%D Leonhard Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 22.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
%D Hugh L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
%D C. Stanley Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134
%H T. D. Noe, <a href="/A002496/b002496.txt">Table of n, a(n) for n = 1..10000</a>
%H Tewodros Amdeberhan, Luis A. Medina and Victor H. Moll, <a href="http://dx.doi.org/10.1016/j.jnt.2007.05.008">Arithmetical properties of a sequence arising from an arctangent sum</a>, J. Numb. Theory, Vol. 128, No. 6 (2008), pp. 18071846, eq. (1.10).
%H William D. Banks, John B. Friedlander, Carl Pomerance and Igor E. Shparlinski, <a href="http://math.dartmouth.edu/~carlp/PDF/banksfinal2.pdf">Multiplicative structure of values of the Euler function</a>, 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. 2947.
%H Paul T. Bateman and Roger A. Horn, <a href="http://dx.doi.org/10.1090/S00255718196201486327">A heuristic asymptotic formula concerning the distribution of prime numbers</a>, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363367.
%H N. A. Carella, <a href="https://arxiv.org/abs/1912.05923">The Euler Polynomial Prime Values Problem</a>, arXiv:1912.05923 [math.GM], 2019.
%H Frank Ellermann, <a href="/A002496/a002496.txt">Primes of the form (m^2)+1 up to 10^6</a>.
%H Leon Mirsky, <a href="http://www.jstor.org/stable/2305811">The number of representations of an integer as the sum of a prime and a kfree integer</a>, Amer. Math. Monthly, Vol. 56, No. 1 (1949), pp. 1719.
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1701.04741">New Congruences and Finite Difference Equations for Generalized Factorial Functions</a>, arXiv:1701.04741 [math.CO], 2017.
%H Apoloniusz Tyszka, <a href="https://philarchive.org/rec/TYSDAS">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)</a>, 2019.
%H Apoloniusz Tyszka and Sławomir Kurpaska, <a href="https://philarchive.org/archive/TYSDASv104">Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X</a>, (2020).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LandausProblems.html">Landau's Problems</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NearSquarePrime.html">NearSquare Prime</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture">BatemanHorn Conjecture</a>.
%H Marek Wolf, <a href="http://arXiv.org/abs/0803.1456">Search for primes of the form m^2+1</a>, arXiv:0803.1456 [math.NT], 20082010.
%F There are O(sqrt(n)/log(n)) members of this sequence up to n. But this is just an upper bound. See the BatemanHorn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.
%F a(n) = 1 + A005574(n)^2.  _R. J. Mathar_, Jul 31 2015
%F Sum_{n>=1} 1/a(n) = A172168.  _Amiram Eldar_, Nov 14 2020
%p select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # _Robert Israel_, Oct 14 2014
%t Select[Range[100]^2+1, PrimeQ]
%t Join[{2},Select[Range[2,300,2]^2+1,PrimeQ]] (* _Harvey P. Dale_, Dec 18 2018 *)
%o (PARI) isA002496(n) = isprime(n) && issquare(n1) \\ _Michael B. Porter_, Mar 21 2010
%o (PARI) is_A002496(n)=issquare(n1)&&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
%o (MAGMA) [p: p in PrimesUpTo(100000) IsSquare(p1)]; // _Vincenzo Librandi_, Apr 09 2011
%o (Haskell)
%o a002496 n = a002496_list !! (n1)
%o a002496_list = filter ((== 1) . a010051') a002522_list
%o  _Reinhard Zumkeller_, May 06 2013
%o (Python)
%o # Python 3.2 or higher required
%o from itertools import accumulate
%o from sympy import isprime
%o A002496_list = [n+1 for n in accumulate(range(10**5),lambda x,y:x+2*y1) if isprime(n+1)] # _Chai Wah Wu_, Sep 23 2014
%o (Python)
%o # Python 2.4 or higher required
%o from sympy import isprime
%o A002496_list = list(filter(isprime, (n*n+1 for n in range(10**5)))) # _David Radcliffe_, Jun 26 2016
%Y Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).
%Y Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293, A172168.
%Y Cf. A000668 (Mersenne primes), A019434 (Fermat primes).
%Y Subsequence of A039770.
%Y Cf. A010051, subsequence of A002522.
%Y Cf. A237040 (an analog for n^3 + 1).
%Y Cf. A010051, A000290; subsequence of A028916.
%Y Subsequence of A039770, A054754, A054755, A063752.
%Y 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).
%Y Cf. A030430 (primes ending with 1), A030432 (primes ending with 7).
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
%E Formula, reference, and comment from _Charles R Greathouse IV_, Aug 24 2009
%E Edited by _M. F. Hasler_, Oct 14 2014
