%I #162 Feb 16 2025 08:32:45

%S 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,

%T 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,

%U 191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277

%N Odd primes.

%C Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers.

%C Odd primes can be written as a sum of no more than two consecutive positive integers. Powers of 2 do not have a representation as a sum of k consecutive positive integers (other than the trivial n=n, for k=1). See A111774. - _Jaap Spies_, Jan 04 2007

%C Intersection of A005408 and A000040. - _Reinhard Zumkeller_, Oct 14 2008

%C Primes which are the sum of two consecutive numbers. - _Juri-Stepan Gerasimov_, Nov 07 2009

%C The arithmetic mean of divisors of p^3, (1+p)(1+p^2)/4, for odd primes p is an integer. - _Ctibor O. Zizka_, Oct 20 2009

%C Primes == -+ 1 (mod 4). - _Juri-Stepan Gerasimov_, Apr 27 2010

%C a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - _Reinhard Zumkeller_, Jul 23 2010

%C Triads of the form <2*a(n+1), a(n+1), 3*a(n+1)> like <6,3,9>, <10,5,15>, <14,7,21> appear in the EKG sequence A064413, see Theorem (3) there. - _Paul Curtz_, Feb 13 2011.

%C Complement of A065090; abs(A151763(a(n))) = 1. - _Reinhard Zumkeller_, Oct 06 2011

%C Right edge of the triangle in A065305. - _Reinhard Zumkeller_, Jan 30 2012

%C Numbers with two odd divisors. - _Omar E. Pol_, Mar 24 2012

%C Odd prime p divides some (2^k + 1) or (2^k - 1), (k>0, minimal, cf. A003558) depending on the parity of A179480((p+1)/2) = r. This is a consequence of the Quasi-order theorem and corollaries, [Hilton and Pederson, pp. 260-264]: 2^k == (-1)^r mod b, b odd; and b divides 2^k - (-1)^r, where p is a subset of b. - _Gary W. Adamson_, Aug 26 2012

%C Subset of the arithmetic numbers (A003601). - _Wesley Ivan Hurt_, Sep 27 2013

%C Odd primes p satisfy the identity: p = (product(2*cos((2*k+1)*Pi/(2*p)), k=0..(p-3)/2))^2. This follows from C(2*p, 0) = (-1)^((p-1)/2)*p, n>=2, with the minimal polynomial C(k,x) of rho(k) := 2*cos(Pi/k). See A187360 for C and the W. Lang link on the field Q(rho(n)), eqs. (20) and (37). - _Wolfdieter Lang_, Oct 23 2013

%C Numbers m > 1 such that m^2 divides (2m-1)!! + m. - _Thomas Ordowski_, Nov 28 2014

%C Numbers m such that m divides 2*(m-3)! + 1. - _Thomas Ordowski_, Jun 20 2015

%C Numbers m such that (2m-3)!! == m (mod m^2). - _Thomas Ordowski_, Jul 24 2016

%C Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - _Thomas Ordowski_, Jul 27 2016

%C Primes of the form x^2 - y^2. - _Thomas Ordowski_, Feb 27 2017

%C Conjecture: a(n) is the smallest odd number m > prime(n) such that Sum_{k=1..prime(n)-1} k^(m-1) == prime(n)-1 (mod m). This is an extension of the Agoh-Giuga conjecture. - _Thomas Ordowski_, Aug 01 2018

%C Numbers k > 1 such that either Phi(k,x) == 1 (mod k) or Phi(k,x) == k (mod k^2) holds, where Phi(k,x) is the k-th cyclotomic polynomial. - _Jianing Song_, Aug 02 2018

%D Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.

%H Ray Chandler, <a href="/A065091/b065091.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harry J. Smith)

%H M. O. Rayes, V. Trevisan and P. S. Wang, <a href="http://icm.mcs.kent.edu/reports/1998/ICM-199802-0001.pdf">Factorization of Chebyshev Polynomials</a>, ICM Report, 1998.

%H M. O. Rayes, V. Trevisan and P. S. Wang, <a href="http://dx.doi.org/10.1016/j.camwa.2005.07.003">Factorization of Chebyshev Polynomials</a>, Computers & Mathematics with Applications, Volume 50, Issues 8-9, October-November 2005, Pages 1231-1240.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeNumber.html">Prime Number</a>.

%F a(n) = A000040(n+1). - _M. F. Hasler_, Oct 26 2013

%p A065091 := proc(n) RETURN(ithprime(n+1)) end:

%t Prime[Range[2, 33]] (* _Vladimir Joseph Stephan Orlovsky_, Aug 22 2008 *)

%o (Haskell)

%o a065091 n = a065091_list !! (n-1)

%o a065091_list = tail a000040_list -- _Reinhard Zumkeller_, Jan 30 2012

%o (Sage)

%o def A065091_list(limit): # after Minác's formula

%o f = 3; P = [f]

%o for n in range(3, limit, 2):

%o if (f+1)>n*(f//n)+1: P.append(n)

%o f = f*n

%o return P

%o A065091_list(100) # _Peter Luschny_, Oct 17 2013

%o (PARI) forprime(p=3, 200, print1(p, ", ")) \\ _Felix Fröhlich_, Jun 30 2014

%o (Magma) [NthPrime(n): n in [2..100]]; // _Vincenzo Librandi_, Jun 21 2015

%o (Python)

%o from sympy import prime

%o def A065091(n): return prime(n+1) # _Chai Wah Wu_, Jul 13 2024

%Y Cf. A000040, A033270, union of A002144 and A002145.

%Y Cf. A230953 (boustrophedon transform).

%K nonn,easy,changed

%O 1,1

%A _Labos Elemer_, Nov 12 2001

%E More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 05 2002

%E Edited (moved contributions from A000040 to here) by _M. F. Hasler_, Oct 26 2013