%I #27 Sep 20 2024 02:04:04
%S 2,4,14,22,28,233,249,261,488,497,511,515,519,526,531,534,548,562,620,
%T 633,635,2985,3119,3123,3128,3157,4350,4358,4392,4438,4474,4484,4606,
%U 4610,4759,5191,12493,1761067,2785124,2785152,2785718,2785729,2867471
%N Numbers k such that 1 + Sum_{i=1..k-1} A001223(i)*(-1)^i = 0.
%C Or, with prime(0) = 1, numbers n such that Sum{k=0..n-1} (prime(k+1)-prime(k))*(-1)^k = Sum{k=0..n-1} (A008578(k+1)-A008578(k))*(-1)^k = 0.
%C There are 313 terms < 10^7, 846 terms < 10^8, 1161 terms < 10^9.
%H Jinyuan Wang, <a href="/A127596/b127596.txt">Table of n, a(n) for n = 1..1161 (first 846 terms from Klaus Brockhaus)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AndricasConjecture.html">Andrica's Conjecture</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeDifferenceFunction.html">Prime Difference Function</a>
%e 1 - A001223(1) = 1 - 1 = 0, hence 2 is a term.
%e 1 - A001223(1) + A001223(2) - A001223(3) = 1 - 1 + 2 - 2 = 0, hence 4 is a term.
%t S=0; Do[j=Prime[n+1]; i=Prime[n]; d[n]=j-i; S=S+(d[n]*(-1)^n); If[S+1==0, Print[Table[j|PrimePi[j]|S+1]]], {n,1,10^7,1}]
%o (PARI) {m=10^8; n=1; p=1; e=1; s=0; while(n<m, q=nextprime(p+1); s=s+(q-p)*e; if(s==0, print1(n, ",")); p=q; e=-e; n++)} /* Klaus Brockhaus, Apr 29 2007 */
%Y Cf. A001223 (differences between consecutive primes), A008578 (prime numbers at the beginning of the 20th century), A000101 (increasing gaps between primes, upper end), A002386 (increasing gaps between primes, lower end).
%Y Cf. A282178 (prime(a(n))), A330545, A330547.
%K nonn
%O 1,1
%A _Manuel Valdivia_, Apr 03 2007
%E Edited by _Klaus Brockhaus_, Apr 29 2007