A098033 Parity of p*(p+1)/2 for n-th prime p.

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Author

Jeremy Gardiner, Sep 10 2004

Keywords

• easy
• nonn

Comments

The following sequences (possibly with a different offset for first term) all appear to have the same parity: A034953 = triangular numbers with prime indices; A054269 = length of period of continued fraction for sqrt(p), p prime; A082749 = difference between the sum of the next prime(n) natural numbers and the sum of the next n primes; A006254 = numbers n such that 2n-1 is prime; A067076 = numbers n such that 2n+3 is a prime.

Analogous to the prime race (mod 3). - Robert G. Wilson v, Sep 17 2004

See also A089253 = 2n-5 is a prime.

For n > 1, if A000040(n) == 1 (mod 4), then a(n) = 1, otherwise a(n)=0, so (for n>1) also a(n) = number of representations of A000040(n) as a difference of hexagonal numbers (A000384) (cf. [Nyblom, p. 262]). - L. Edson Jeffery, Feb 16 2013

Examples

			a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Crossrefs

Cf. A034953, A054269, A082749, A006254, A067076, A100672.

Programs

Maple

seq((ithprime(n) mod 4) mod 3, n = 2..105] # Gary Detlefs, Oct 27 2011

Mathematica

Table[ Mod[ Prime[n](Prime[n] + 1)/2, 2], {n, 105}] (* Robert G. Wilson v, Sep 17 2004 *)
Mod[(#(#+1))/2,2]&/@Prime[Range[110]] (* Harvey P. Dale, Mar 29 2015 *)

PARI

a(n)=prime(n)%4<3 \\ Charles R Greathouse IV, Oct 27 2011

Formula

a(n) = parity of p*(p+1)/2 for n-th prime p.

a(n) = 1 - A100672(n), n > 1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008

For n > 1, a(n) = (prime(n) mod 4) mod 3. - Gary Detlefs, Oct 27 2011

Extensions

More terms from Robert G. Wilson v, Sep 17 2004