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A065091 Odd primes. 121
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

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

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 *)

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

CROSSREFS

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

Cf. A230953 (boustrophedon transform).

Sequence in context: A176997 A006005 * A160656 A240699 A065380 A211075

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 September 17 09:38 EDT 2014. Contains 246836 sequences.