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A098033 Parity of p(p+1)/2 for n-th prime p. 1
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 (list; graph; refs; listen; history; text; internal format)
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 next prime(n) natural numbers and the sum of 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

Table of n, a(n) for n=1..105.

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(n)

a(n) = (p(n) mod 4) mod 3, n>1 for n-th prime p(n). - 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.

equal to 1 minus A100672. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008

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

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)