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

1,1

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

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.

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A295892 A120522 A157423 * A284471 A135022 A286726

Adjacent sequences: A098030 A098031 A098032 * A098034 A098035 A098036

Keyword

easy,nonn

Author

Jeremy Gardiner, Sep 10 2004

Extensions

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

Status

approved