A267084 - OEIS

%I #23 Sep 25 2017 13:06:52

%S 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N a(n) = ceiling(A007504(n)/n) - floor(A007504(n)/n); a(n) is 0 if n divides the sum of first n primes, 1 otherwise.

%C a(n) = 0 for n=1, 23, 53, 853, ... see A045345.

%C It is conjectured that there are infinitely many zeros, but that their density is zero.

%H Antti Karttunen, <a href="/A267084/b267084.txt">Table of n, a(n) for n = 1..12000</a>

%H Javier Cilleruelo and Florian Luca, <a href="http://digital.csic.es/bitstream/10261/31070/1/Sum%2520of%2520primes.pdf">On the sum of the first n primes</a>, Q. J. Math. 59:4 (2008), 14 pp.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = A225804(n) - A060620(n).

%t Table[Ceiling[(Plus@@Prime[Range[n]])/n]-Floor[(Plus@@Prime[Range[n]])/n], {n, 100}]

%o (PARI)

%o up_to = 105

%o v007504 = vector(up_to,i,prime(i));

%o for(i=2,up_to,v007504[i] = v007504[i-1]+v007504[i]); \\ Taking partial sums of primes here.

%o A007504(n) = v007504[n];

%o A267084(n) = if(!(A007504(n)%n),0,1); \\ _Antti Karttunen_, Sep 24 2017

%o (Scheme) (define (A267084 n) (if (zero? (modulo (A007504 n) n)) 0 1)) ;; _Antti Karttunen_, Sep 24 2017

%Y Cf. A007504, A045345 (positions of zeros), A060620, A158682, A225804.

%K nonn,easy

%O 1

%A _Ctibor O. Zizka_, Jan 10 2016

%E More terms and the second description added to the name by _Antti Karttunen_, Sep 24 2017