|
| |
| |
|
|
|
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
|
|
|
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
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 *)
|
|
|
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: A073579 A006005 * A160656 A176997 A240699 A065380
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
|
| |
|
|
