This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A098033 #45 Jan 05 2025 19:51:37
%S A098033 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,
%T A098033 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,
%U A098033 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
%N A098033 Parity of p*(p+1)/2 for n-th prime p.
%C A098033 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.
%C A098033 Analogous to the prime race (mod 3). - _Robert G. Wilson v_, Sep 17 2004
%C A098033 See also A089253 = 2n-5 is a prime.
%C A098033 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
%H A098033 Amiram Eldar, <a href="/A098033/b098033.txt">Table of n, a(n) for n = 1..10000</a>
%H A098033 M. A. Nyblom, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-3/nyblom.pdf">On the representation of the integers as a difference of nonconsecutive triangular numbers</a>, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
%F A098033 a(n) = parity of p*(p+1)/2 for n-th prime p.
%F A098033 a(n) = 1 - A100672(n), n > 1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008
%F A098033 For n > 1, a(n) = (prime(n) mod 4) mod 3. - _Gary Detlefs_, Oct 27 2011
%e A098033 a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).
%p A098033 seq((ithprime(n) mod 4) mod 3, n = 2..105] # _Gary Detlefs_, Oct 27 2011
%t A098033 Table[ Mod[ Prime[n](Prime[n] + 1)/2, 2], {n, 105}] (* _Robert G. Wilson v_, Sep 17 2004 *)
%t A098033 Mod[(#(#+1))/2,2]&/@Prime[Range[110]] (* _Harvey P. Dale_, Mar 29 2015 *)
%o A098033 (PARI) a(n)=prime(n)%4<3 \\ _Charles R Greathouse IV_, Oct 27 2011
%Y A098033 Cf. A034953, A054269, A082749, A006254, A067076, A100672.
%K A098033 easy,nonn
%O A098033 1,1
%A A098033 _Jeremy Gardiner_, Sep 10 2004
%E A098033 More terms from _Robert G. Wilson v_, Sep 17 2004