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 A065091 Odd primes. 225
 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Rayes et al. prove that the a(n)-th Chebyshev-T polynomial, divided by x, is irreducible over the integers. 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 Intersection of A005408 and A000040. - Reinhard Zumkeller, Oct 14 2008 Primes which are the sum of two consecutive numbers. - Juri-Stepan Gerasimov, Nov 07 2009 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 Primes == -+ 1 (mod 4). - Juri-Stepan Gerasimov, Apr 27 2010 a(n) = A053670(A179675(n)) and a(n) <> A053670(m) for m < A179675(n). - Reinhard Zumkeller, Jul 23 2010 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. Complement of A065090; abs(A151763(a(n))) = 1. - Reinhard Zumkeller, Oct 06 2011 Right edge of the triangle in A065305. - Reinhard Zumkeller, Jan 30 2012 Numbers with two odd divisors. - Omar E. Pol, Mar 24 2012 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 Subset of the arithmetic numbers (A003601). - Wesley Ivan Hurt, Sep 27 2013 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 Numbers m > 1 such that m^2 divides (2m-1)!! + m. - Thomas Ordowski, Nov 28 2014 Numbers m such that m divides 2*(m-3)! + 1. - Thomas Ordowski, Jun 20 2015 Numbers m such that (2m-3)!! == m (mod m^2). - Thomas Ordowski, Jul 24 2016 Odd numbers m such that ((m-3)!!)^2 == +-1 (mod m). - Thomas Ordowski, Jul 27 2016 Primes of the form x^2 - y^2. - Thomas Ordowski, Feb 27 2017 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 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 REFERENCES Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106. LINKS Harry J. Smith, Table of n, a(n) for n = 1..1000 M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, ICM Report, 1998. M. O. Rayes, V. Trevisan and P. S. Wang, Factorization of Chebyshev Polynomials, Computers & Mathematics with Applications, Volume 50, Issues 8-9, October-November 2005, Pages 1231-1240. Eric Weisstein's World of Mathematics, Prime Number. FORMULA a(n) = A000040(n+1). - M. F. Hasler, Oct 26 2013 MAPLE A065091 := proc(n) RETURN(ithprime(n+1)) end: MATHEMATICA Prime[Range[2, 33]] (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *) NestList[NextPrime@# &, 3, 60] (* Robert G. Wilson v, Aug 02 2018 *) PROG (PARI) { for (n=1, 1000, write("b065091.txt", n, " ", prime(n + 1)) ) } \\ Harry J. Smith, Oct 06 2009 (PARI) The program below is supposedly valid for generating primes for n>=3; it is based on the comment in A075888: "For n>=3, prime(n+1)^2-prime(n)^2 is always divisible by 24" j=[]; for(n=0, 500, if((floor(sqrt(4!*(n+1) + 1))) == ceil(sqrt(4!*(n+1) + 1)), if(isprime(floor(sqrt(4!*(n+1) + 1))), j=concat(j, floor(sqrt(4!*(n+1) + 1)))))); j \\ Alexander R. Povolotsky, Sep 16 2008 (Haskell) a065091 n = a065091_list !! (n-1) a065091_list = tail a000040_list  -- Reinhard Zumkeller, Jan 30 2012 (Sage) def A065091_list(limit):  # after Minác's formula     f = 3; P = [f]     for n in range(3, limit, 2):         if (f+1)>n*(f//n)+1: P.append(n)         f = f*n     return P A065091_list(100)  # Peter Luschny, Oct 17 2013 (PARI) forprime(p=3, 200, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014 (MAGMA) [NthPrime(n): n in [2..100]]; // Vincenzo Librandi, Jun 21 2015 CROSSREFS Cf. A000040, A033270, union of A002144 and A002145. Cf. A230953 (boustrophedon transform). Sequence in context: A175524 A073579 A006005 * A160656 A176997 A240699 Adjacent sequences:  A065088 A065089 A065090 * A065092 A065093 A065094 KEYWORD nonn,easy AUTHOR Labos Elemer, Nov 12 2001 EXTENSIONS More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Jan 05 2002 Edited (moved contributions from A000040 to here) by M. F. Hasler, Oct 26 2013 STATUS approved

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Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)