A065091 Odd primes.

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

Offset: 1

Author

Labos Elemer, Nov 12 2001

Keywords

• nonn
• easy

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

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
• 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.

Crossrefs

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

Cf. A230953 (boustrophedon transform).

Programs

Haskell

a065091 n = a065091_list !! (n-1)
a065091_list = tail a000040_list  -- Reinhard Zumkeller, Jan 30 2012

Magma

[NthPrime(n): n in [2..100]]; // Vincenzo Librandi, Jun 21 2015

Maple

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

Mathematica

Prime[Range[2, 33]] (* Vladimir Joseph Stephan Orlovsky, Aug 22 2008 *)

PARI

forprime(p=3, 200, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

Python

from sympy import prime
def A065091(n): return prime(n+1) # Chai Wah Wu, Jul 13 2024

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

Formula

a(n) = A000040(n+1). - M. F. Hasler, Oct 26 2013

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